Preduals of JBW*-triples are 1-Plichko spaces

Preduals of JBW*-triples are 1-Plichko spaces Abstract We investigate the preduals of JBW*-triples from the point of view of Banach space theory. We show that the algebraic structure of a JBW*-triple M naturally yields a decomposition of its predual M*, by showing that M* is a 1-Plichko space (that is, it admits a countably 1-norming Markushevich basis). In case M is σ-finite, its predual M* is even weakly compactly generated. These results are a common roof for previous results on L1-spaces, preduals of von Neumann algebras, and preduals of JBW*-algebras. 1. Introduction The topic of this paper concerns the interplay between operator algebras, Jordan structures and Banach space theory. The main goal is to prove that the predual of any JBW*-triple satisfies the remarkable Banach space feature called 1-Plichko property (see Theorem 1.1 below). The predual of a JBW*-triple can be viewed as a non-commutative and non-associative generalization of an L1 space. In general such a space may be highly non-separable. Despite this fact, our result implies that the predual of a JBW*-triple admits a nice decomposition into separable subspaces and admits an appropriate Markushevich basis. More precisely, let X be a Banach space. A subspace D⊂X* is said to be a Σ-subspace of X* if there is a linearly dense set S⊂X such that D={ϕ∈X*:{m∈S:ϕ(m)≠0}iscountable}. The Banach space X is called ( r-)Plichko if X* admits an ( r-)norming Σ-subspace, that is there exists a Σ-subspace D of X* such that ∥x∥≤rsup{∣ϕ(x)∣:ϕ∈D,∥ϕ∥≤1}(x∈X) (compare [34, 37]). We prove that the predual M* of any JBW*-triple M is 1-Plichko by identifying a 1-norming Σ-subspace of M=(M*)*. Moreover, the Σ-subspace we find is a canonical one and has an easy algebraic description (see Section 4) and it is even an inner ideal (see Theorem 5.1). This witnesses a close relationship of the algebraic and Banach space structures of JBW*-triples. The 1-Plichko property of a Banach space X is equivalent to the fact that X has a countably 1-norming Markushevich basis [34, Lemma 4.19]. Another deep result [41, Theorem 27] says that X is a 1-Plichko space if and only if it admits a commutative 1-projectional skeleton. A commutative 1-projectional skeleton is a system (Pλ)λ∈Λ of norm one projections on X, where Λ is an up-directed set, fulfilling the following conditions: PλX is separable for each λ and X=⋃λ∈ΛPλX. PλPμ=Pλ whenever λ≤μ. PλPμ=PμPλ for all λ and μ. If (λn) is an increasing net in Λ, it has a supremum, λ∈Λ, and PλX=⋃nPλnX¯. It easily follows that any 1-Plichko space enjoys the 1-separable complementation property saying that any separable subspace can be enlarged to a 1-complemented separable subspace. This property was established by U. Haagerup for preduals of von Neumann algebras with the help of results from modular theory of von Neumann algebras (see [26, Theorem IX.1]). The category of 1-Plichko spaces involves many classes of spaces studied in Banach space theory. Let us recall that X is weakly Lindelöf determined (WLD), if X* is a Σ-subspace of itself. X is called weakly compactly generated (WCG) if it contains a weakly compact subset whose linear span is dense in X. Obviously, every WLD space is 1-Plichko, and it follows from [1, Proposition 2] that every WCG space is WLD. Plichko and 1-Plichko spaces were formally introduced in [34, Section 4.2]. The notion was motivated by a series of papers where A. N. Plichko studied this property under equivalent reformulations (see [48–51]). Although the term 1-Plichko is the most commonly used name for the spaces defined above, they have been also studied under different names. Namely, the class of those Banach spaces which are 1-Plichko is precisely the class termed V by J. Orihuela [44], which has been also studied by M. Valdivia [56]. It has been proved by the third author of this note in [37] that many important spaces have 1-Plichko property, for example L1 spaces for non-negative σ-finite measures, order-continuous Banach lattices, and C(K)-spaces for abelian compact groups K. Moreover, the paper [37] contains the first result on non-commutative L1 spaces showing that the predual of a semi-finite von Neumann algebra is 1-Plichko. Motivated by the latter, the first three authors of this paper prove in [4] that the predual of any von Neumann algebra is 1-Plichko. Moreover, they showed that the canonical 1-norming Σ-subspace is the two-sided ideal of all elements whose range projection is σ-finite. A generalization to JBW*-algebras appeared to be non-trivial. In [5] the same authors showed that the predual of any JBW*-algebra is 1-Plichko. The proof was quite different from that given in the setting of von Neumann algebras. The proof in the Jordan case was based on constructing a special projection skeleton with the help of the set theoretical tool of elementary submodels. Obviously, the question whether, as in the case of von Neumann algebra preduals [4], the result can be obtained without any use of submodels theory is a gap which is not fulfilled by the results in [5]. In the present paper, we prove a further generalization of the above mentioned results by showing that all JBW*-triple preduals are 1-Plichko spaces. Our main result reads as follows. Theorem 1.1. The predual M*of a JBW*-triple Mis a 1-Plichko space. Moreover, M*is WLD if and only if Mis σ-finite. In this case, M* WCG. The approach in this paper resembles more the one of [4] than the one of [5]. One reason for this has already been mentioned, in the present paper the proofs and arguments do not make use of the set theoretic tool of submodels. Moreover, the theory of JBW*-triples allows to connect the description of the Σ-subspace obtained in [4] and to obtain a similar and satisfactory description for JBW*-triples (and hence also for JBW*-algebras), see Theorem 5.1. The key result for this approach is Proposition 4.3. The relevant notions related to JBW*-triples are gathered in Section 2. Theorem 1.1—in fact a more precise version of Theorem 1.1—follows from Theorems 3.1 and 4.1 proved below. Since the second dual of a JB*-triple is a JBW*-triple (see [10, Corollary 3.3.5]), the next result is a straightforward consequence of Theorem 1.1. Corollary 1.2. The dual space of a JB*-triple is a 1-Plichko space. We recall that a Banach space X has the ( r-)separable complementation property if any separable subspace of X is contained in an ( r-)complemented separable subspace of X (compare [26, page 92]). Since 1-Plichko spaces enjoy the 1-separable complementation property (which follows immediately from the characterization using a projectional skeleton formulated above), we also get the following result. Corollary 1.3. Preduals of JBW*-triples have the 1-separable complementation property. The above corollary is an extension of a result of U. Haagerup, who showed that the same statement holds for von Neumann algebra preduals (with different methods, see [26, Theorem IX.1]). 2. Notation and preliminaries In this section, we recall basic notions and results on JBW*-triples and Plichko spaces. We also include some auxiliary results needed to prove our main results. For unexplained notation from Banach space theory, we refer to [21]. The symbols BX and X* will denote the closed unit ball and the dual of a Banach space X, respectively. 2.1. Elements of JBW*-triples W. Kaup [39] obtains an analytic-algebraic characterization of bounded symmetric domains in terms of the so-called JB*-triples, by showing that every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB*-triple. Thanks to this result, JB*-triples offer a natural bridge to connect the infinite-dimensional holomorphy with functional analysis. We recall that a JB*-triple is a complex Banach space E equipped with a continuous ternary product {.,.,.}, which is symmetric and bilinear in the outer variables and conjugate-linear in the middle one, satisfying the following properties: {x,y,{a,b,c}}={{x,y,a},b,c}−{a,{y,x,b},c}+{a,b,{x,y,c}} for all a,b,c,x,y∈E (Jordan identity), the operator x↦{a,a,x} is a Hermitian operator with non-negative spectrum for each a∈E, ∥{a,a,a}∥=∥a∥3 for a∈E. We recall that an operator T∈B(E) is Hermitian if and only if ∥exp(irT)∥=1 for each r∈R. For a,b∈E we define a (linear) operator L(a,b) on E by L(a,b)(x)={a,b,x}, x∈E, and a conjugate-linear operator Q(a,b) by Q(a,b)(x)={a,x,b}. Given a∈E, the symbol Q(a) will denote the operator on E defined by Q(a)=Q(a,a). Every C*-algebra is a JB*-triple with respect to the triple product given by {x,y,z}=12(xy*z+zy*x). The same triple product equips the space B(H,K), of all bounded linear operators between complex Hilbert spaces H and K, with a structure of JB*-triple. Among the examples involving Jordan algebras, we can say that every JB*-algebra is a JB*-triple under the triple product {x,y,z}=(x◦y*)◦z+(z◦y*)◦x−(x◦z)◦y*. An element e in a JB*-triple E is said to be a tripotent if e={e,e,e}. If E is a von Neumann algebra viewed as a JBW*-triple, then any projection is clearly a tripotent; in fact, an element of a von Neumann algebra is a tripotent if and only if it is a partial isometry. For each tripotent e∈E, the mappings Pi(e):E→E(i=0,1,2) defined by P2(e)=L(e,e)(2L(e,e)−idE),P1(e)=4L(e,e)(idE−L(e,e))andP0(e)=(idE−L(e,e))(idE−2L(e,e)) are contractive linear projections (see [23, Corollary 1.2]), called the Peirce projections associated with e. It is known (cf. [10, p. 32]) that P2(e)=Q(e)2, P1(e)=2(L(e,e)−Q(e)2), and P0(e)=idE−2L(e,e)+Q(e)2. In case E is a von Neumann algebra, e∈E a partial isometry, q=e*e the initial projection and p=ee* the final projection, we get P2(e)x=pxq,P1(e)x=px(1−q)+(1−p)xqandP0(e)x=(1−p)x(1−q). If e is even a symmetric element (that is, e*=e) in the von Neumann algebra then we have p=q. The range of Pi(e) is the eigenspace, Ei(e), of L(e,e) corresponding to the eigenvalue i2, and E=E2(e)⊕E1(e)⊕E0(e) is termed the Peirce decomposition of E relative to e. Clearly, e∈E2(e) and Pk(e)(e)=0 for k=0,1. The following multiplication rules (known as Peirce rules or Peirce arithmetic) are satisfied: {E2(e),E0(e),E}={E0(e),E2(e),E}={0}, (2.1) {Ei(e),Ej(e),Ek(e)}⊆Ei−j+k(e), (2.2) where Ei−j+k(e)={0} whenever i−j+k∉{0,1,2} ([23] or [10, Theorem 1.2.44]). A tripotent e is called complete if E0(e)={0}. The complete tripotents of a JB*-triple E are precisely the complex and the real extreme points of its closed unit ball (cf. [6, Lemma 4.1] and [38, Proposition 3.5] or [10, Theorem 3.2.3]). Therefore every JBW*-triple contains an abundant collection of complete tripotents. If E=E2(e), or equivalently, if {e,e,x}=x for all x∈E, we say that e is unitary. For each tripotent e in a JB*-triple, E, the Peirce-2 subspace E2(e) is a unital JB*-algebra with unit e, product a◦eb≔{a,e,b} and involution a⁎e≔{e,a,e} (cf. [10, Section 1.2 and Remark 3.2.2]). As we noticed above, every JB*-algebra is a JB*-triple with respect to the product {a,b,c}=(a◦b*)◦c+(c◦b*)◦a−(a◦c)◦b*. Kaup’s Banach–Stone theorem (see [39, Proposition 5.5]) assures that a surjective operator between JB*-triples is an isometry if and only if it is a triple isomorphism. Consequently, the triple product on E2(e) is uniquely determined by the expression {a,b,c}=(a◦eb⁎e)◦ec+(c◦eb⁎e)◦ea−(a◦ec)◦b⁎e, (2.3) for every a,b,c∈E2(e). Therefore, unital JB*-algebras are in one-to-one correspondence with JB*-triples admitting a unitary element (see also [9, 4.1.55]). A JBW*-triple is a JB*-triple which is also a dual Banach space. Examples of JBW*-triples include von Neumann algebras and JBW*-algebras. Every JBW*-triple admits a unique isometric predual and its triple product is separately weak*-to- weak*-continuous ([3, 29], [10, Theorem 3.3.9]). Consequently, the Peirce projections associated with a tripotent in a JBW*-triple are weak*-to- weak*-continuous. Therefore, for each tripotent e in a JBW*-triple M, the Peirce subspace M2(e) is a JBW*-algebra. Unlike general JB*-triples, JBW*-triples admit a rather concrete representation which we recall in Section 2.4 below as it is the essential tool for proving our results. Let a,b be elements in a JB*-triple E. Following standard terminology, we shall say that a and b are algebraically orthogonal or simply orthogonal (written a⊥b) if L(a,b)=0. If we consider a C*-algebra A as a JB*-triple, then two elements a,b∈A are orthogonal in the C*-sense (that is, ab*=b*a=0) if and only if they are orthogonal in the triple sense. Orthogonality is a symmetric relation. By Peirce arithmetic it is immediate that all elements in E2(e) are orthogonal to all elements in E0(e), in particular, two tripotents u,v∈E are orthogonal if and only if u∈E0(v) (and, by symmetry, if and only if v∈E0(u)). We refer to [8, Lemma 1] for other useful characterizations of orthogonality and additional details not explained here. The order in the partially ordered set of all tripotents in a JB*-triple E is defined as follows: Given two tripotents e,u∈E, we say that e≤u if u−e is a tripotent which is orthogonal to e. Lemma 2.1. ([10, 23, Corollary 1.7, Proposition 1.2.43]). Let u,ebe two tripotents in a JB*-triple E. The following assertions are equivalent. e≤u. P2(e)(u)=e. {u,e,u}=e. {e,u,e}=e. e is a projection (that is, a self-adjoint idempotent) in the JB*-algebra E2(u). For each norm-one functional φ in the predual M*, of a JBW*-triple M, there exists a unique tripotent e∈M satisfying φ=φP2(e) and φ∣M2(e) is a faithful normal state of the JBW*-algebra M2(e) (see [23, Proposition 2]). This unique tripotent e is called the support tripotent of φ, and will be denoted by e(φ). It is explicitly shown in [23] that ifuisatripotentinMwith1=∥φ∥=φ(u),thenu≥e(φ). (2.4) We recall that a subspace I of a JB*-triple E is called an inner ideal, provided {I,E,I}⊆I (that is, provided {a,b,c}∈I whenever a,c∈I and b∈E, see [16]). Clearly, an inner ideal is a subtriple. Note that if e is a tripotent of a JBW*-triple M, then M2(e) is a weak*-closed inner ideal of M(compare the previous Peirce arithmetic). In a von Neumann algebra W (regarded as JBW*-triple) left and right ideals and sets of the form aWb (with fixed a,b∈W) are inner ideals, whereas weak*-closed inner ideals are of the form pWq with projections p,q∈W [15, Theorem 3.16]. Given an element x in a JB*-triple E, the symbol Ex will denote the norm-closed subtriple of E generated by x, that is, the closed subspace generated by all odd powers x[2n+1], where x[1]=x, x[3]={x,x,x}, and x[2n+1]={x,x,x[2n−1]} ( n≥2) (compare, for example, [42, Section 3.3] or [10, Lemma 1.2.10]). It is known that there exists an isometric triple isomorphism Ψ:Ex→C0(L) satisfying Ψ(x)(t)=t, for all t in L (compare [39, 1.15]), where C0(L) is the abelian C*-algebra of all complex-valued continuous functions on L vanishing at 0, L being a locally compact subset of (0,∥x∥] satisfying that L∪{0} is compact. Thus, for any continuous function f:L∪{0}→C vanishing at 0, it is possible to give the usual meaning in the sense of functional calculus to f(x)∈Ex, via f(x)=Ψ−1(f). For each norm-one element x in a JBW*-triple M, r(x) will denote its range tripotent. We succinctly describe its definition. (More details are given for example in [46, Section 2.2] or in [14, comments before Lemma 3.1] or [8, Section 2]). For x∈M with ∥x∥=1, the functions t→t12n−1 give rise to an increasing sequence (x[12n−1]) which weak*-converges to r(x) in M. The tripotent r(x) is the smallest tripotent e∈M satisfying that x is a positive element in the JBW*-algebra M2(e) (see, for example, [14, comments before Lemma 3.1] or [8, Section 2]). The inequality x≤r(x) holds in M2(r(x)) for every norm-one element x∈E. For a non-zero element z∈M, the range tripotent of z, r(z), is precisely the range tripotent of z∥z∥, and we set r(0)=0. Let M be a JBW*-triple. We recall that a tripotent u in M is said to be σ-finite if u does not majorize an uncountable orthogonal subset of tripotents in M. Equivalently, u is a σ-finite tripotent in M if and only if there exists an element φ in M* whose support tripotent e(φ) coincides with u (cf. [17, Theorem 3.2]). Following standard notation, we shall say that M is σ-finite if every tripotent in M is σ-finite, equivalently, every orthogonal subset of tripotents in M is countable (cf. [17, Proposition 3.1]). It is also known that the sum of an orthogonal countable family of mutually orthogonal σ-finite tripotents in M is again a σ-finite tripotent (see [17, Theorem 3.4(i)]). It is further proved in [17, Theorem 3.4(ii)] that every tripotent in M is the supremum of a set of mutually orthogonal σ-finite tripotents in M. When a von Neumann algebra W is regarded as a JBW*-triple, a projection p is σ-finite in the triple sense if and only if it is σ-finite or countably decomposable in the usual sense employed for von Neumann algebras (compare [53, Definition 2.1.8] or [55, Definition II.3.18]). We will need the following properties of σ-finite tripotents which have been borrowed from [17]. Lemma 2.2. ([17]). Let Mbe a JBW*-triple and let ebe a tripotent of M. Then the following hold: M2(e)is a JBW*-subtriple of Mand any tripotent p∈M2(e)is σ-finite in M2(e)if and only if it is σ-finite in M. eis σ-finite if and only if M2(e)is σ-finite. If eis σ-finite, then any tripotent in M2(e)is σ-finite in M. Proof Since M2(e) is a weak*-closed subtriple of M, assertion (i) follows from [17, Lemma 3.6(ii)]. Assertion (ii) follows from (i), [17, Theorem 4.4(viii)–(ix)] and the fact that e is a complete tripotent in M2(e). Finally, assertion (iii) follows immediately from (i) and (ii).□ For non-explained notions concerning JB*-algebras and JB*-triples, we refer to the monographs [9, 10]. 2.2. Contractive and bicontractive projections One of the main properties enjoyed by any member E in the class of JB*-triples states that the image of a contractive projection P:E→E (where contractive means ∥P∥≤1) is again a JB*-triple with triple product {x,y,z}P≔P({x,y,z}) for x,y,z in P(E) and P{a,x,b}=P{a,P(x),b},a,b∈P(E),x∈E (2.5) (see [24, 35, 49]). It is further known that under these conditions P(E) need not be, in general, a JB*-subtriple of E (compare [22, Example 1] or [40, Example 3]). But note that if P(E) is known to be a subtriple, then {·,·,·}P coincides with the original triple product of E because in JB*-triples norm and triple product determine each other (see e.g. [10, Theorem 3.1.7, 3.1.20]). Fortunately, more can be said about the JB*-triple structure of P(E). It is known that P(E) is isometrically isomorphic to a JB*-subtriple of E** (see [25, Theorem 2]). If P:E→E is even a bicontractive projection (where bicontractive means ∥P∥≤1 and ∥I−P∥≤1—by IV or simply I we denote the identity on a vector space V) on a JB*-triple, it satisfies a stronger property. Namely, P(E) is then a JB*-subtriple of E, in particular (2.5) can be improved because the identities P{a,b,x}={a,b,P(x)}andP{a,x,b}={a,P(x),b} (2.6) hold for a,b∈P(E), x∈E (cf. [25, Section 3]). It is further known that when P is bicontractive, there exists a surjective linear isometry (that is, a triple automorphism) Θ on E of period 2 such that P=12(I+Θ) (see [25, Theorem 4]). Since, by another interesting property of JBW*-triples, every surjective linear isometry on a JBW*-triple is weak*-to- weak*-continuous (see [29, Proof of Theorem 3.2]) we have, as a consequence, that a bicontractive projection P on a JBW*-triple is weak*-to- weak*-continuous. 2.3. Von Neumann tensor products We recall now some basic facts on von Neumann tensor products of von Neumann algebras. The theory has been essentially borrowed from [55, Chapter IV], and we refer to the latter monograph for additional results not commented here. Let A⊂B(H) and W⊂B(K) be von Neumann algebras. The algebraic tensor product A⊗W is canonically embedded into B(H⊗K), where H⊗K is the Hilbertian tensor product of H and K (see [55, Definition IV.1.2]). The von Neumann algebra generated by the algebraic tensor product A⊗W is denoted A⊗¯W, and is called the von Neumann tensor product of A and W. Note that A⊗¯W is the weak* closure of A⊗W in B(H⊗K) (see [55, Section IV.5]). If A is commutative, then the predual of A⊗¯W is canonically identified with the projective tensor product of preduals, that is (A⊗¯W)*=A*⊗^πW*. (2.7) This follows from [55, Theorem IV.7.17] (or rather [55, Section IV.7]). Furthermore, the special case of a separable W* is treated in [53, Theorem 1.22.13], while there is another approach via results on operator spaces: Results due to E. G. Effros and Z. J. Ruan show that equality (2.7) holds for any von Neumann algebra W, when the projective tensor product on the right-hand side is in the category of operator spaces ([18, Theorem 7.2.4], [19]). But if A is commutative, it carries the minimal operator-space structure by [18, Proposition 3.3.1], and hence the predual A* carries the maximal structure by [18, (3.3.13) or (3.3.15) on p. 51], and hence by [18, (8.2.4) on p. 146] the projective tensor product in the category of operator spaces coincides with the projective tensor product in the Banach space sense. Lemma 2.3. Let Aand Wbe von Neumann algebras with Acommutative. Suppose P:W→Wis a weak*-to- weak*-continuous contractive projection. Then the following holds: P(W)is a JBW*-triple with triple product {x,y,z}P≔P({x,y,z})for x,y,zin P(W). A⊗¯P(W), the weak*-closure of the algebraic tensor product A⊗P(W)in A⊗¯W, is the range of a weak*-to- weak*-continuous contractive projection Q on A⊗¯W. A⊗¯P(W)is a JBW*-triple with triple product {x,y,z}Q≔Q({x,y,z})for x,y,zin A⊗¯P(W). Moreover, (A⊗¯P(W))*=A*⊗^π(P(W))*=A*⊗^πP*(W*). Proof We know from Section 2.2 that statement (i) is satisfied. Since P is weak*-to- weak* continuous, it is the dual map of a map P*:W*→W*. It is clear that P* is a contractive projection on W*. It follows from basic tensor product properties (cf. [11, 3.2] or [52, Proposition 2.3]) that I⊗P* is a contractive projection on A*⊗^πW*. Moreover, by [11, 3.8] or [52, Proposition 2.5] the norm on its range (which is the norm-closure of the algebraic tensor product A*⊗P*(W*)) is the projective norm coming from A*⊗^πP*(W*). Further, it is clear that the dual mapping Q=(I⊗P*)* is a weak*-to- weak*-continuous contractive projection on (A*⊗^πW*)*=A⊗¯W. Using the results commented in Section 2.2 we know that its range is a JBW*-triple with the triple product defined in (iii). Since the range of Q is canonically identified with the dual of A*⊗^πP*(W*), to complete the proof of (ii) and (iii) it is enough to show that the range of (I⊗P*)* is A⊗¯P(W). To show the desired equality we observe that the restriction of (I⊗P*)* to the algebraic tensor product A⊗W coincides with I⊗P. Therefore the range of (I⊗P*)* contains A⊗P(W) and hence also its weak* closure A⊗¯P(W). Conversely, since the unit ball BA⊗W is weak*-dense in BA⊗¯W (for example by the Kaplansky density theorem), and (I⊗P*)* is weak*-to- weak*-continuous, BA⊗W is weak* dense in the unit ball of the range of (I⊗P*)* as well. This completes the proof.□ Lemma 2.4. Let Aand Wbe von Neumann algebras with Acommutative. Suppose P:W→Wis a bicontractive projection. Then the following holds: P(W)is a JBW*-subtriple of W. A⊗¯P(W), the weak*-closure of the algebraic tensor product A⊗P(W)in A⊗¯W, is the range of a bicontractive projection on A⊗¯W. A⊗¯P(W)is a JBW*-subtriple of A⊗¯Wand, moreover, (A⊗¯P(W))*=A*⊗^π(P(W))*=A*⊗^πP*(W*). Proof By Section 2.2, we know that P(W) is a JB*-subtriple of W and that P is weak*-to- weak*-continuous. Hence we can apply Lemma 2.3. Moreover, since P is even bicontractive, we get that P* is bicontractive, and hence I⊗P* and Q=(I⊗P*)* are bicontractive too. Finally, since Q is bicontractive, by Section 2.2 we get that A⊗¯P(W) is a JBW*-subtriple of A⊗¯W.□ 2.4. Structure theory In this subsection, we recall an important structure result, due to G. Horn [30] and G. Horn and E. Neher [31], which allows us to represent every JBW*-triple in a concrete way. These results will be the main tool for proving that JBW*-triple preduals are 1-Plichko spaces. We begin by recalling the definition of Cartan factors. There are six types of them (compare [10, Example 2.5.31]): Type 1: A Cartan factor of type 1 coincides with a Banach space B(H,K), of all bounded linear operators between two complex Hilbert spaces H and K, where the triple product is defined by {x,y,z}=2−1(xy*z+zy*x). If dimH=dimK, then we can suppose H=K and we get the von-Neumann algebra B(H). If dimK<dimH, we may suppose that K is a closed subspace of H and then B(H,K) is a JB*-subtriple of B(H). Moreover, if p is the orthogonal projection of H onto K, then x↦px is a bicontractive projection of B(H) onto B(H,K). If dimK>dimH, we may suppose that H is a closed subspace of K, p the orthogonal projection of K onto H and then x↦xp is a bicontractive projection of B(K) onto B(H,K). Types 2 and 3: Cartan factors of types 2 and 3 are the subtriples of B(H) defined by C2={x∈B(H):x=−jx*j} and C3={x∈B(H):x=jx*j}, respectively, where j is a conjugation (that is, a conjugate-linear isometry of period 2) on H. If j is a conjugation on H, then there is an orthonormal basis (eγ)γ∈Γ such that j(∑γ∈Γcγeγ)=∑γ∈Γcγ¯eγ. Each x∈B(H) can be represented by a ‘matrix’ (xγδ)γ,δ∈Γ. It is easy to check that the representing matrix of jx*j is the transpose of the representing matrix of x. Hence, C2 consists of operators with antisymmetric representing matrix and C3 of operators with symmetric ones. Therefore, P(x)=12(xt+x) (where xt=jx*j is the transpose of x with respect to the basis chosen above) is a bicontractive projection on B(H) such that C3 is the range of P, and C2 is the range of I−P. Type 4: A Cartan factor of type 4 (denoted by C4) is a complex spin factor, that is, a complex Hilbert space (with inner product ⟨.,.⟩) provided with a conjugation x↦x¯, triple product {x,y,z}=⟨x,y⟩z+⟨z,y⟩x−⟨x,z¯⟩y¯, and norm given by ∥x∥2=⟨x,x⟩+⟨x,x⟩2−∣⟨x,x¯⟩∣2. We point out that C4 is isomorphic to a Hilbert space and hence, in particular, reflexive. Types 5 and 6: All we need to know about Cartan factors of types 5 and 6 (also called exceptional Cartan factors) is that they are all finite dimensional. Although H. Hanche-Olsen showed in [27, Section 5] that the standard method to define tensor products of JC-algebras (and JW *-triples) is, in general, hopeless, von Neumann tensor products can be applied in the representation theory of JBW *-triples. Let A be a commutative von Neumann algebra and let C be a Cartan factor which can be realized as a JW*-subtriple of some B(H). As before, the symbol A⊗¯C will denote the weak*-closure of the algebraic tensor product A⊗C in the usual von Neumann tensor product A⊗¯B(H) of A and B(H). This applies to Cartan factors of types 1–4 (this is obvious for Cartan factors of types 1–3, the case of type 4 Cartan factors follows from [28, Theorem 6.2.3]). The above construction does not cover Cartan factors of types 5 and 6. When C is an exceptional Cartan factor, A⊗¯C will denote the injective tensor product of A and C, which can be identified with the space C(Ω,C), of all continuous functions on Ω with values in C endowed with the pointwise operations and the supremum norm, where Ω denotes the spectrum of A (cf. [52, p. 49]). We observe that if C is a finite dimensional Cartan factor which can be realized as a JW*-subtriple of some B(H) both definitions above give the same object (cf. [55, Theorem IV.4.14]). The structure theory settled by G. Horn and E. Neher [30, 31, (1.7)] proves that every JBW*-triple M writes (uniquely up to triple isomorphisms) in the form M=(⨁j∈JAj⊗¯Cj)ℓ∞⊕ℓ∞H(W,α)⊕ℓ∞pV, (2.8) where each Aj is a commutative von Neumann algebra, each Cj is a Cartan factor, W and V are continuous von Neumann algebras, p is a projection in V, α is a linear involution on W commuting with *, that is, a linear *-antiautomorphism of period 2 on W, and H(W,α)={x∈W:α(x)=x}. 2.5. Some facts on Plichko spaces The following lemma sums up several basic properties of Σ-subspaces. Lemma 2.5. Let Xbe a Banach space and D⊂X*a Σ-subspace. Then the following hold: Dis weak*-countably closed. That is, C¯w*⊂Dwhenever C⊂Dis countable. In particular, Dis weak*-sequentially closed and norm-closed. Bounded subsets of Dare weak*-Fréchet-Urysohn. That is, given A⊂Dbounded and x*∈Dsuch that x*∈A¯w*, then there is a sequence (xn*)in A weak*-converging to x*. Let D′⊂X*be any other subspace satisfying (i) and (ii). If D∩D′is 1-norming, then D=D′. If Xis WLD, then X*is the only norming Σ-subspace of X*. If Dis 1-norming, then for any x∈Xthere is x*∈Dof norm one such that x*(x)=∥x∥. Proof Assertion (i) follows from the very definition of a Σ-subspace, assertion (ii) follows from [34, Lemma 1.6]. Assertion (iii) is an easy consequence of (i) and (ii) and follows from [35, Lemma 2] (in fact in the just quoted lemma it is assumed that D′ is a Σ-subspace as well, but the proof uses only properties (i) and (ii)). Assertion (iv) follows immediately from (iii) and (v) is an easy consequence of (i).□ We will also need the following easy lemma on quotients of 1-Plichko spaces. Lemma 2.6. Let Xbe a 1-Plichko Banach space, and let D⊂X*be a 1-norming Σ-subspace. Suppose that Z⊂X*is a weak*-closed subspace such that D∩BZis weak*dense in BZ. Then D∩Zis a 1-norming Σ-subspace of Z=(X/Z⊥)*. Proof Since Z is a weak*-closed subspace of the dual space X*, it is canonically isometrically identified with (X/Z⊥)*. Further, by the assumptions it is clear that D∩Z is a 1-norming subspace of Z. It remains to show it is a Σ-subspace. To do that, fix a linearly dense set S⊂X such that D={x*∈X*:{x∈S:x*(x)≠0}iscountable}. Let S˜ be the image of S in X/Z⊥ by the canonical quotient mapping. It is clear that S˜ is linearly dense. Let D˜={x*∈Z=(X/Z⊥)*:{x∈S˜:x*(x)≠0}iscountable} be the Σ-subspace induced by S˜. It is easy to check that D∩Z⊂D˜. It follows from Lemma 2.5(iii) that D∩Z=D˜, which completes the proof.□ 3. Preduals of σ-finite JBW*-triples The aim of this section is to prove the following result. Theorem 3.1. The predual of any σ-finite JBW*-triple is WCG, in fact even Hilbert-generated. Recall that a Banach space X is said to be Hilbert-generated if there is a Hilbert space H and a bounded linear mapping T:H→X with dense range. It is clear that any Hilbert-generated Banach space is WCG (the generating weakly compact set is precisely T(BH)). Theorem 3.1 above follows from the following stronger statement, which is a JBW*-triple analog of [4, Lemma 3.3] for von Neumann algebras and of [5, Proposition 3.7] for JBW*-algebras. Proposition 3.2. Let ebe a σ-finite tripotent in a JBW*-triple M. Then the predual of the space M2(e)⊕M1(e) (i.e. (P2(e)+P1(e))*(M*)) is Hilbert-generated. To see that Theorem 3.1 follows from the above proposition it is enough to use the fact that any JBW*-triple contains an abundant set of complete tripotents. In particular, any σ-finite JBW*-triple M contains a σ-finite complete tripotent e∈M such that M=M2(e)⊕M1(e). Hence Proposition 3.2 entails Theorem 3.1. Next let us focus on the proof of Proposition 3.2. Similarly as in the case of von Neumann algebras and JBW*-algebras, it will be done by introducing a canonical (semi)definite inner product. Barton and Friedman [2, Proposition 1.2] showed that given an element φ in the dual of a JB*-triple E and an element z∈E such that φ(z)=∥φ∥=∥z∥=1, the map E×E∋(x,y)↦⟨x,y⟩φ≔φ{x,y,z} defines a hermitian semi-positive sesquilinear form with the associated pre-hilbertian seminorm ∥x∥φ≔(φ{x,x,z})1/2 on M and is independent of z. We shall need the following technical lemma borrowed from [17, Lemma 4.1]: Lemma 3.3. Let Mbe a JBW*-triple, let φ∈M*be of norm one and let e=e(φ)∈Mbe its support tripotent. Then the annihilator of the pre-Hilbertian seminorm ∥·∥φis precisely M0(e), that is, {x∈M:∥x∥φ=0}=M0(e). (3.1)In particular, the restriction of ∥·∥φto M2(e)⊕M1(e)is a pre-Hilbertian norm and the restriction of ⟨·,·⟩φto M2(e)⊕M1(e)is an inner product. Proof The first statement is proved in [17, Lemma 4.1], the positive definiteness of ∥·∥φ and of ⟨·,·⟩φ on M2(e)⊕M1(e) follows immediately (see also [23, Lemma 1.5], [45]).□ Now we are ready to prove the main proposition of this section: Proof of Proposition 3.2 Since e is a σ-finite tripotent, there exists a norm-one normal functional φ∈M* such that e=e(φ) is the support tripotent of φ. Denote by hφ the pre-Hilbertian space M2(e)⊕M1(e) equipped with the inner product ⟨·,·⟩φ=φ{·,·,e}, and write Hφ for its completion. Let us first consider Φ˜(a) defined by x↦⟨x,a⟩φ for a∈hφ, x∈M. By the Cauchy-Schwarz inequality we have ∣Φ˜(a)(x)∣=∣⟨x,a⟩φ∣≤∥x∥φ∥a∥φ≤∥x∥∥a∥φ which, together with the separate w*-continuity of the triple product, shows that Φ˜ is a well-defined conjugate-linear contractive map from hφ to M*. In order to see that the range of Φ˜ is contained in (M2(e)⊕M1(e))*=(P2*(e)+P1*(e))(M*), let us observe that for any a∈hφ and y∈M0(e), we have ∥y∥φ=0 by Lemma 3.3, and hence Φ˜(a)(y)=0. Thus, by density of hφ in Hφ, Φ˜=(P2*(e)+P1*(e))Φ˜ gives rise to a conjugate-linear continuous map Φ:Hφ→(M2(e)⊕M1(e))*. We shall finally prove that Φ has norm-dense range. Suppose z∈M2(e)⊕M1(e) satisfies Φ(a)(z)=0 for every a∈hφ. In particular, 0=Φ(z)(z)=∥z∥φ2 and thus, by Lemma 3.3, z=0. By the Hahn-Banach theorem, Φ has dense range. If we replace the map Φ by Φj, where j is a conjugation on Hφ, then we have a linear mapping.□ 4. The case of general JBW*-triples In this section, we state and prove Theorem 4.1, which gives a more precise version of the first part of Theorem 1.1. To provide a precise formulation, we introduce one more notation. For a JBW* triple M we define the set Mσ={x∈M:thereisaσ-finitetripotente∈MsuchthatP2(e)x=x} and note that Mσ={x∈M:thereisaσ-finitetripotente∈Msuchthat{e,e,x}=x}={x∈M:r(x)isaσ-finitetripotent}. Indeed, the first equality follows from the fact that the range of P2(e) is the eigenspace of L(e,e) corresponding to the eigenvalue 1. Let us show the second equality. The inclusion ‘ ⊃’ is obvious. To show the converse inclusion, let x∈Mσ. Fix a σ-finite tripotent e∈M with x=P2(e)x, that is, x∈M2(e). Since M2(e) is a JBW*-subtriple of M and r(x) belongs to the JBW*-subtriple generated by x, we have r(x)∈M2(e) and so r(x) is σ-finite by Lemma 2.2. We mention the easy but useful fact that Mσ is 1-norming in M=(M*)*. To see this we simply observe that Mσ contains all σ-finite tripotents of M, or equivalently, all support tripotents of functionals in M*. Theorem 4.1. The predual space of a JBW*-triple Mis a 1-Plichko space. Moreover, Mσisa1-normingΣ-subspaceofM=(M*)*. (4.1)In particular, M*is WLD if and only if Mis σ-finite. It is not obvious that Mσ is a subspace, but this will follow by the proof of Theorem 4.1; it will be reproved a second time in Theorem 5.1. The ‘in particular’ part of the theorem is an immediate consequence of the first statements of the theorem. Indeed, M is σ-finite if and only if M=Mσ (cf. Lemma 2.2). Hence, if M is σ-finite, then M* is WLD by the first statement. Conversely, if M* is WLD, then by the first part of the theorem together with Lemma 2.5 (iv) we get M=Mσ, hence M is σ-finite. Thus, it is enough to prove (4.1). This will be done in the rest of this section by using results in [4] and the decomposition (2.8). The following proposition is almost immediate from the main results of [4]. Proposition 4.2. The statement of Theorem4.1holds for von Neumann algebras. Proof It is enough to show (4.1) in case M is a von Neumann algebra. In view of [4, Proposition 4.1], to this end it is enough to show that Mσ={x∈M:x=qxqforaσ-finiteprojectionq∈M}. Let x be in the set on the right-hand side. Fix a σ-finite projection q∈M with x=qxq. Then q is a σ-finite tripotent and {q,q,x}=12(qx+xq)=qxq=x. Hence x∈Mσ. Conversely, let x∈Mσ and let u∈M be a σ-finite triponent with x=P2(u)x. Since M is a von Neumann algebra, u is a partial isometry and hence P2(u)x=pxq, where p=uu* is the final projection and q=u*u is the initial projection. Then p is a σ-finite projection. Indeed, suppose that (rγ)γ∈Γ is an uncountable family of pairwise orthogonal projections smaller than p. Then it is easy to check that (rγu)γ∈Γ is an uncountable family of pairwise orthogonal tripotents smaller than u. Similarly we get that q is σ-finite. Hence their supremum r=p∨q is σ-finite as well ([17, Theorem 3.4] or [33, Exercice 5.7.45]) and satisfies x=rxr. Thus x belongs to the set on the right-hand side and the proof is complete.□ Proposition 4.3. Let P:M→Mbe a bicontractive projection on a JBW*-triple, let N=P(M), and let ebe a tripotent in N. Then eis σ-finite in Nif and only if eis σ-finite in M, that is, Nσ=N∩Mσ. Proof The ‘if’ implication is clear. Let e be a σ-finite tripotent in N. By [17, Theorem 3.2] there exists a norm-one functional ϕ∈N* whose support tripotent in N is e. Let us define ψ=P*(ϕ)=ϕP∈M*. Clearly ∥ψ∥=1. We shall prove that e is the support tripotent of ψ in M, and hence e is σ-finite in M ([17, Theorem 3.2]). Let u be the support tripotent of ψ in M. From ψ(e)=ϕ(e)=1=∥ψ∥ we get e≥u (compare [23, part (b) in the proof of Proposition 2]). We set u1=P(u) and u2=u−u1. Since e≥u in M, we deduce that {e,u,e}=u={e,e,u} ( e−u∈M0(u) and Peirce rules). Hence, u1=P(u)={e,Pu,e}={e,u1,e} and u1={e,e,u1} by (2.6). It follows that u1={e,u1,e}∈M2(e) and that u1={e,u1,e}=u1⁎e is a hermitian element in the closed unit ball of the JBW*-algebra N2(e). As e is the unit in this algebra and u1 is a hermitian element of norm less than one, we see that e−u1 is a positive element in the JBW*-algebra N2(e). The condition ϕ(e)=1=ψ(u)=ϕP(u)=ϕ(u1) implies, by the faithfulness of ϕ∣N2(e), that u1=e. It follows from the above that u2={e,e,u}−{e,e,u1}={e,e,u2} and similarly u2={e,u2,e}. These identities combined with the fact that u=e+u2 is a tripotent (that is, {e+u2,e+u2,e+u2}=e+u2) yield e+u2=e+2{u2,u2,e}+{u2,e,u2}+3u2+{u2,u2,u2}. After applying the bicontractive projection I−P in both terms of the last equality we get −2u2={u2,u2,u2}. Now 2∥u2∥=∥{u2,u2,u2}∥=∥u2∥3 implies either u2=0 or ∥u2∥2=2. The latter is not possible because ∥u2∥≤1 by the fact that u2=(I−P)u and I−P is a contraction. Thus u2=0, and hence e=u, which proves the first statement. For the last identity, we observe that for every element x∈N, its range tripotent r(x) (in N or in M) lies in N. Suppose x is an element in N whose range tripotent is σ-finite in N. We deduce from the first statement that r(x) is also σ-finite in M, and hence Nσ⊆Mσ. The inclusion Nσ⊇Mσ∩N is clear.□ By combining Proposition 4.2, Proposition 4.3 and Lemma 2.6 we get the following proposition. Proposition 4.4. Let P:W→Wbe a bicontractive projection on a von Neumann algebra W, let M=P(W). Then M*is a 1-Plichko space. Furthermore, Mσis a 1-norming Σ-subspace of M. Now we are ready to prove the validity of (4.1) for most of the summands from the representation (2.8): Proposition 4.5. Let Mbe a JBW*-triple of one of the following forms: M=A⊗¯C, where Ais a commutative von Neumann algebra and Cis a Cartan factor of type 1, 2 or3. M=H(W,α), where Wis a von Neumann algebra and αis a linear involution on Wcommuting with *. M=pV, where Vis a von Neumann algebra and p∈Vis a projection. Then Mσ is a 1-norming Σ-subspace of M=(M*)*. Proof We will apply Proposition 4.4. To do that it is enough to show that M is the range of a bicontractive projection on a von Neumann algebra. (a) If C is a Cartan factor of type 1, 2 or 3, then C is the range of a bicontractive projection on a certain von Neumann algebra W, as it was previously observed after the definitions of the respective Cartan factors. The desired bicontractive projection on A⊗¯W is finally given by Lemma 2.4. (b) A bicontractive projection on W is given by x↦12(x+α(x)). (c) The mapping x↦px defines a bicontractive projection on V.□ The remaining summands from (2.8) are covered by the following theorem, which we formulate in a more abstract setting of Banach spaces. Theorem 4.6. Let (Ω,Σ,μ)be a measure space with a non-negative semifinite measure, and let Ebe a reflexive Banach space. Then the space L1(μ,E)of Bochner-integrable functions is 1-Plichko. Furthermore, L1(μ,E)is WLD if and only if μis σ-finite, in the latter case it is even WCG. More precisely, there is a family of finite measures (μγ)γ∈Γsuch that L1(μ,E)is isometric to (⨁γ∈ΓL1(μγ,E))ℓ1and D={f=(fγ)γ∈Γ∈(⨁γ∈ΓL∞(μγ,E))ℓ∞:{γ∈Γ:fγ≠0}iscountable}is a 1-norming Σ-subspace of (L1(μ,E))*=(⨁γ∈ΓL∞(μγ,E))ℓ∞. Proposition 4.7. Let μbe a finite measure, and let Ebe a reflexive Banach space. Then L1(μ,E)is WCG. Proof The proof is done similarly as in the scalar case (cf. [37, Theorem 5.1]). Let us consider the identity mapping T:L2(μ,E)→L1(μ,E). By the Cauchy-Schwarz inequality we get ∥T∥≤∥μ∥, hence T is a bounded linear operator. Moreover, the range of T is dense, since countably valued functions in L1(μ,E) are dense in the latter space. Finally, L2(μ,E) is reflexive because E and E* have Radon-Nikodým property (see [12, Theorem IV.1.1]). Thus, L1(μ,E) is indeed WCG.□Remark: Note that if E is isomorphic to a Hilbert space, then we can even conclude that L1(μ,E) is Hilbert-generated, since in this case L2(μ,E) is also isomorphic to a Hilbert space. Indeed, if E is even isometric to a Hilbert space, the norm on L2(μ,E) is induced by the scalar product ⟨f,g⟩=∫⟨f(ω),g(ω)⟩dμ(ω). Proof of Theorem 4.6 We imitate the proof of [37, Theorem 5.1]. Let B⊂Σ be a maximal family with the following properties: 0<μ(B)<+∞ for each B∈B; μ(B1∩B2)=0 for each B1,B2∈B distinct. The existence of such a family follows immediately from Zorn’s lemma. Take any separable-valued Σ-measurable function f:Ω→E. Then clearly ∫∥f(ω)∥dμ(ω)=∑B∈B∫B∥f(ω)∥dμ(ω). Therefore, L1(μ,E) is isometric to the ℓ1-sum of spaces L1(μ∣B,E), B∈B. Since μ∣B is finite for each B∈B, L1(μ∣B,E) is WCG (and hence WLD) by the previous Proposition 4.7. Further, it is clear that the dual of L1(μ,E) is canonically isometric to the ℓ∞-sum of the family {(L1(μ∣B,E))*:B∈B}. More concretely, since E is reflexive, by [12, Theorem IV.1.1] we have (L1(μ∣B,E))*=L∞(μ∣B,E*) for each B∈B, and hence L1(μ,E)*=(⨁B∈BL∞(μ∣B,E*))ℓ∞. Finally, it follows from [34, Lemma 4.34] that D={(fB)B∈B∈(⨁B∈BL∞(μ∣B,E*))ℓ∞:{B∈B;fB≠0}iscountable} is a 1-norming Σ-subspace of (L1(μ,E))*. To prove the last statement, it is enough to observe that μ is σ-finite if and only if B is countable, that a countable ℓ1-sum of WCG spaces is again WCG and that an uncountable ℓ1-sum of nontrivial spaces contains ℓ1(ω1) and hence is not WLD. (Recall that WLD property passes to subspaces.)□ Proposition 4.8. Let Abe a commutative von Neumann algebra and Ca Cartan factor. Then (A⊗¯C)*=A*⊗^πC*. Proof If C is a Cartan factor of type 1, 2 or 3, then C is the range of a bicontractive projection on a von Neumann algebra and hence the equality follows from Lemma 2.4. If C is a type 4 Cartan factor, it follows from [20, Lemma 2.3] that C is the range of a (unital positive) contractive projection P:B(H)→B(H) where H is an appropriate Hilbert space. The mapping P**:B(H)**→B(H)** is a weak*-to- weak*-continuous contractive projection on the von Neumann algebra B(H)** whose range is C by (Goldstine’s theorem and) reflexivity of C. Hence the desired equality follows from Lemma 2.3. If C is a Cartan factor of type 5 or 6, then it is finite dimensional and A⊗¯C is defined to be the injective tensor product. Further, by [11, 3.2] or [52, p. 24] we get (A*⊗^πC*)*=B(A*,C) which coincides with the injective tensor product A⊗^εC, as C has finite dimension.□ Lemma 4.9. Let (Mγ)γ∈Γbe an indexed family of JBW*-triples, and let us denote M=(⨁γ∈ΓMγ)ℓ∞. Then Mσ={(xγ)γ∈Γ∈M:xγ∈(Mγ)σforγ∈Γand{γ∈Γ:xγ≠0}iscountable}. Proof This follows easily if we observe that e=(eγ)γ∈Γ∈M is a tripotent if and only if eγ is a tripotent for each γ and, moreover, e is σ-finite if and only if each eγ is σ-finite and only countably many eγ are non-zero.□ Proposition 4.10. Let Abe a commutative von Neumann algebra and Ca reflexive Cartan factor. (This applies, in particular, to Cartan factors of types 4, 5 and 6.) Let M=A⊗¯C. Then Mσis a 1-norming Σ-subspace of M=(M*)*, and hence M*is 1-Plichko. Furthemore, M*is WLD if and only if Ais σ-finite. In such a case M*is even WCG. Proof If A is a commutative von Neumann algebra, by [55, Theorem III.1.18] it can be represented as L∞(Ω,μ), where Ω is a locally compact space and μ a positive Radon measure on Ω. In fact, Ω is the topological sum of a family of compact spaces (Kγ)γ∈Γ. Then the predual of A is identified with L1(Ω,μ)=(⨁γ∈ΓL1(Kγ,μ∣Kγ))ℓ1. Since (A⊗¯C)*=A*⊗^πC*=L1(μ,C*), we can use Theorem 4.6. To complete the proof it is enough to show that D=Mσ, where D is the Σ-subspace provided by Theorem 4.6. Since M=(⨁γ∈ΓL∞(Kγ,μ∣Kγ,C))ℓ∞, due to Lemma 4.9, it is enough to show that L∞(μ,C) is σ-finite whenever μ is finite. But, in this case, its predual, L1(μ,C*), is WCG by Proposition 4.7, thus L∞(μ,C) is σ-finite by Theorem 4.6.□ Proof of Theorem 4.1 We have already mentioned that it is enough to show (4.1). Let M be a JBW*-triple and consider the decomposition (2.8). By Propositions 4.5 and 4.10 each summand fulfills (4.1). Further, Lemma 4.9 and [34, Lemma 4.34] yield the validity of (4.1) for M.□ In passing we remark that from Theorem 4.1 (and the general facts on Plichko spaces), we have that Mσ is norm-closed and even weak*-countably closed; it is additionally weak*-closed if and only if M is σ-finite. 5. Structure of the space Mσ In the previous section we proved that, for any JBW*-triple M, Mσ is a 1-norming Σ-subspace of M=(M*)*. If M is σ-finite, it is the only 1-norming Σ-subspace and coincides with the whole M. If M is not σ-finite, there may be plenty of different 1-norming Σ-subspaces (cf. [34, Example 6.9]). However, Mσ is the only canonical 1-norming Σ-subspace. What we mean by this statement is in the content of the following theorem. Theorem 5.1. Let Mbe a JBW*-triple. Then Mσis a norm-closed inner ideal in M. Moreover, it is the only 1-norming Σ-subspace which is also an inner ideal. The theorem will be proved at the end of this section. The following technical result provides a characterization of σ-finite tripotents which is required later. We recall that, given a tripotent u in a JBW*-triple M, there exists a complete tripotent w∈M such that u≤w (see [29, Lemma 3.12(1)]). Proposition 5.2. Let ube a tripotent in a JBW*-triple M. The following statements are equivalent: uis σ-finite; There exist a σ-finite tripotent vand a complete tripotent win Msuch that v≤wand (w−v)⊥u. Proof The implication (a)⇒(b) is clear with v=u and any complete tripotent w in M with u≤w . (b)⇒(a) Suppose there exist a σ-finite tripotent v and a complete tripotent w in M such that v≤w and (w−v)⊥u. Writing w=v+(w−v) and using successively the orthogonality of w−v to u and to v we obtain {w,w,u}={w,v,u}={v,v,u}, and hence L(w,w)u=L(v,v)u, and similarly {w,u,w}={v,u,v}. Since w−v⊥M2(v)∋{v,u,v}, it follows that P2(w)(u)=Q(w)2(u)={w,{v,u,v},w}={v,{v,u,v},v}=P2(v)(u).Therefore, P2(w)(u)=P2(v)(u) and P1(w)(u)=2L(w,w)(u)−2P2(w)(u)=P1(v)(u). The completeness of w assures that u=P2(w)(u)+P1(w)(u)=P2(v)(u)+P1(v)(u) lies in M2(v)⊕M1(v). We shall show now that u is σ-finite. Arguing by contradiction, assume there is an uncountable family (uj)j∈Γ of mutually orthogonal non-zero tripotents in M with uj≤u for every j (see [17, Section 3]). Since uj∈M2(u) for every j and u⊥(w−v), it follows that uj⊥(w−v) for every j∈Γ. Arguing as above we obtain uj∈M2(v)⊕M1(v), for every j∈Γ. Having in mind that v is σ-finite, we can find a norm one functional ϕv∈M* whose support tripotent is v (see [17, Theorem 3.2]). By Lemma 3.3, ϕv gives rise to a norm ∥·∥ϕv on M2(v)⊕M1(v) defined by ∥x∥ϕv=(ϕv{x,x,v})1/2 ( x∈M2(v)⊕M1(v)). As uj is a non-zero element in M2(v)⊕M1(v) by the preceding paragraph, we obtain ϕv{uj,uj,v}=∥uj∥2>0. Therefore, there exists a positive constant Θ and an uncountable subset Γ′⊆Γ such that ϕv{uj,uj,v}>Θ for all j∈Γ′. Thus, for each natural m we can find j1≠j2≠⋯≠jm∈Γ′. Since the elements uj1,…,ujm are mutually orthogonal, we get 1=∥∑k=1mujk∥2≥∥∑k=1mujk∥ϕv2=ϕv{∑k=1mujk,∑k=1mujk,v}=∑k=1mϕv{ujk,ujk,v}>mΘ, which is impossible.□ To prove that Mσ is an inner ideal, we need another representation of M. To this end fix a complete tripotent e∈M. Applying [17, Theorem 3.4(ii)] we can find a family (eλ)λ∈Λ of mutually orthogonal σ-finite tripotents in M satisfying e=∑λ∈Λeλ. For each x∈M let us define Λx≔{λ∈Λ:L(eλ,eλ)(x)≠0}. Proposition 5.3. In the conditions above, Mσ={x∈M:Λxiscountable}and Mσis a norm-closed inner ideal of M. Proof Denote the set on the right-hand side by Mσ′. By the linearity of the Jordan product in the third variable, it follows that Mσ′ is a linear subspace. To show that it is an inner ideal, take x,z∈Mσ′ and y∈M. For each λ∈Λ⧹(Λx∪Λz), we deduce via Jordan identity, that L(eλ,eλ){x,y,z}={L(eλ,eλ)x,y,z}−{x,L(eλ,eλ)y,z}+{x,y,L(eλ,eλ)z}=−{x,L(eλ,eλ)y,z}. Moreover, since L(eλ,eλ)x=L(eλ,eλ)z=0, we get x,z∈M0(eλ). Since P0(eλ)y is in the 0-eigenspace of L(eλ,eλ) we have that L(eλ,eλ)(y)∈M1(eλ)⊕M2(eλ) and hence {x,L(eλ,eλ)(y),z}=0 by Peirce arithmetic. We have shown that Λ{x,y,z}⊆Λx∪Λz, and thus Λ{x,y,z} is countable, which proves that {x,y,z}∈Mσ′ and hence Mσ′ is an inner ideal of M. We continue by showing that Mσ⊂Mσ′. We shall first prove that Mσ′ contains all σ-finite tripotents in M. Let u be a σ-finite tripotent in M. We want to show that the set Λu is countable. We assume, on the contrary, that Λu is uncountable. Let ϕu∈M* be a norm one functional whose support tripotent is u. For every λ∈Λu, we have that eλ∈M0(u) because otherwise we would have L(eλ,eλ)(u)=0. Consequently, as in the proof of Proposition 5.2, we deduce that ϕu{eλ,eλ,u}>0. We can thus find a positive constant Θ and an uncountable subset Λu′⊆Λu such that ϕu{eλ,eλ,u}>Θ for all λ∈Λu′. As before, for each natural m we can find λ1≠λ2≠⋯≠λm∈Λu′. Then, applying the orthogonality of the elements eλj we get 1=∥∑j=1meλj∥2≥∥∑j=1meλj∥ϕu2=ϕu{∑j=1meλj,∑j=1meλj,u}=∑j=1mϕu{eλj,eλj,u}>mΘ, which gives a contradiction. This proves that Λu is countable, and hence u∈Mσ′. Let us now assume that x is any element of Mσ. Then its range tripotent, r(x), is σ-finite and hence r(x)∈Mσ′ by the previous paragraph. Since x∈M2(r(x)) is a positive and hence self-adjoint element, we have {r(x),x,r(x)}=x and hence x∈Mσ′ as Mσ′ is an inner ideal. This shows that Mσ⊂Mσ′. Conversely, let x∈Mσ′. In this case the set Λx is countable. The tripotent u=w*-∑λ∈Λxeλ is σ-finite in M, e=u+v, where v=w*-∑λ∈Λ⧹Λxeλ is another tripotent in M with u⊥v. Since {eλ,eλ,x}=0 for all λ∈Λ⧹Λx, it follows from the separate weak*-continuity of the triple product of M that {v,v,x}=0, that is, x∈M0(v). Hence also r(x)∈M0(v) (as M0(v) is a JBW*-subtriple of M). It follows that r(x)⊥v and hence r(x) is σ-finite by Proposition 5.2. We finally observe that, by Theorem 4.1, Mσ is a Σ-subspace and hence it is norm-closed (cf. Lemma 2.5(i)). This completes the proof.□ We are now ready to prove the main theorem of this section. Proof of Theorem 5.1 Mσ is a norm-closed inner ideal by Proposition 5.3. Let us prove the uniqueness. Let I be an inner ideal which is a 1-norming Σ-subspace. We will show that I contains all sigma-finite tripotents. Let e∈M be a sigma-finite tripotent, ϕ∈M* a normal functional of norm 1 such that e is the support tripotent of ϕ. By Lemma 2.5(v) there is x∈I of norm 1 with ϕ(x)=1. Further, we get r(x)∈I. Indeed, r(x) is contained in the weak*-closure of the JB*-subtriple of M generated by x. Since this subtriple is norm-separable, we get r(x)∈I by Lemma 2.5(i). In order to show e∈I, it is enough to show that e≤r(x). By (2.4), it is enough to prove that ϕ(r(x))=1. Proposition 2.5 in [45] assures that ϕ(x[12n+1])=ϕ(x)[12n+1]=1, for all natural n. Since ϕ is a normal functional and (x[12n+1])→r(x) in the weak* topology of M, it follows that ϕ(r(x))=1, as we desired. Now, if z∈Mσ is arbitrary, then there is a σ-finite tripotent f∈M with z∈M2(f). By the above we have f∈I. Since I is an inner ideal, we conclude that M2(f)⊂I, and hence z∈I. Therefore, Mσ⊂I. Lemma 2.5(iii) now shows that Mσ=I.□ Remark 5.4. It is possible to give a shorter proof of the fact that the predual of a JBW*-triple is 1-Plichko by using the main result of [5] at the cost of applying elementary submodels theory. However, this alternative argument does not yield Mσ as a concrete description of a Σ-subspace. We shall only sketch this variant: First, it is not too difficult to modify the decomposition (2.8) by writing M=(⨁j∈IAj⊗¯Gj)ℓ∞⊕ℓ∞N⊕ℓ∞pV, (5.1) where each Aj is a commutative von Neumann algebra, each Gj is a finite dimensional Cartan factor, p is a projection in a von Neumann algebra V, and N is a JBW*-algebra. Second, an almost word-by-word adaptation of the proof of [4, Theorem 1.1] shows that the predual of pV is 1-Plichko (compare Proposition 4.5). So is the predual of N by the main result of [5]. Finally, the summands Aj⊗¯Gj are seen to have 1-Plichko predual as in the proof of 4.6 (or by an easier argument using the finite dimensionality of Cj), and the stability of 1-Plichko spaces by ℓ1-sums ([34, Theorem 4.31(iii)] or Lemma 4.9) allows us to conclude. 6. The case of real JBW*-triples Introduced by J. M. Isidro et al. (see [32]), real JB*-triples are, by definition, the closed real subtriples of JB*-triples. Every complex JB*-triple is a real JB*-triple when we consider the underlying real Banach structure. Real and complex C*-algebras belong to the class of real JB*-triples. An equivalent reformulation asserts that real JB*-triples are in one-to-one correspondence with the real forms of JB*-triples. More precisely, for each real JB*-triple E, there exist a (complex) JB*-triple Ec and a period-2 conjugate-linear isometry (and hence a conjugate-linear triple isomorphism) τ:Ec→Ec such that E={b∈Ec:τ(b)=b}. The JB*-triple Ec identifies with the complexification of E (see [32, Proposition 2.2] or [9, Proposition 4.2.54]). In particular, every JB-algebra (and hence the self-adjoint part, Asa of every C*-algebra A) is a real JB*-triple. Henceforth, for each complex Banach space X, the symbol XR will denote the underlying real Banach space. In the conditions above, we can consider another period-2 conjugate-linear isometry τ♯:Ec*→Ec* defined by τ♯(φ)(z)≔φ(τ(z))¯(φ∈Ec*). It is further known that the operator (Ec*)τ♯→(Ecτ)*,φ↦φ∣E is an isometric real-linear bijection, where (Ec*)τ♯≔{φ∈Ec*:τ♯(φ)=φ}. A real JBW*-triple is a real JB*-triple which is also a dual Banach space ([32, Definition 4.1] and [43, Theorem 2.11]). It is known that every real JBW*-triple admits a unique (isometric) predual and its triple product is separately weak*-continuous (see [43, Proposition 2.3 and Theorem 2.11]). Actually, by the just quoted results, given a real JBW*-triple N there exists a JBW*-triple M and a weak*-to- weak* continuous period-2 conjugate-linear isometry τ:M→M such that N=Mτ. The mapping τ♯ maps M* into itself, and hence we can identify (M*)τ♯ with N*=(Mτ)*. We can also consider a weak*-continuous real-linear bicontractive projection P=12(Id+τ) of M onto N=Mτ, and a bicontractive real-linear projection of M* onto N* defined by Q=12(Id+τ♯). From now on, N, M, τ, P and Q will have the meaning explained in this paragraph. Due to the general lack for real JBW*-triples of the kind of structure results established by Horn and Neher for JBW*-triples in [30, 31], the proofs given in Section 4 cannot be applied for real JBW*-triples. Despite of the limitations appearing in the real setting, we shall see how the tools in previous section can be applied to prove that preduals of real JBW*-triples are 1-Plichko spaces too. We shall need to extend the concept of σ-finite tripotents to the setting of real JBW*-triples. The notions of tripotents, Peirce projections, Peirce decomposition are perfectly transferred to the real setting. The relations of orthogonality and order also make sense in the set of tripotents in N (cf. [32, 43]). Furthermore, for each tripotent e in N, Q(e) induces a decomposition of N into R-linear subspaces satisfying N=N1(e)⊕N0(e)⊕N−1(e), where Nk(e)≔{x∈N:Q(e)x=kx}, N2(e)=N1(e)⊕N−1(e)N0(e)=N1(e)⊕N0(e), {Nj(e),Nk(e),Nℓ(e)}⊂Njkℓ(e)ifjkℓ≠0,j,k,ℓ∈{0,±1},andzerootherwise. The natural projection of N onto Nk(e) is denoted by Pk(e). It is also known that P1(e), P−1(e), and P0(e) are all weak*-continuous. The subspace N1(e) is a weak*-closed Jordan subalgebra of the JBW-algebra (M2(e))sa, and hence N1(e) is a JBW-algebra. Given a normal functional ϕ∈N*, there exists a normal functional φ∈M* satisfying τ♯(φ)=φ and φ∣N=ϕ. Let e(φ) be the support tripotent of φ in M. Since 1=φ(e(φ))=φ(τ(e(φ)))¯=φ(τ(e(φ))), we deduce that τ(e(φ))≥e(φ). Applying that τ is a triple homomorphism, we get e(φ)=τ2(e(φ))≥τ(e(φ))≥e(φ), which proves that e(φ)=τ(e(φ))∈N. That is, the support tripotent of a τ♯-symmetric normal functional φ in M* is τ-symmetric. The tripotent e(φ) is called the support tripotent of ϕ in N, and it is denoted by e(ϕ). It is known that ϕ=ϕP1(e(ϕ)) and ϕ∣N1(e(ϕ)) is a faithful positive normal functional on the JBW-algebra N1(e(ϕ)) (compare [47, Lemma 2.7]). As in the complex setting, a tripotent e in N is called σ-finite if e does not majorize an uncountable orthogonal subset of tripotents in N. The real JBW*-triple N is called σ-finite if every tripotent in N is σ-finite. Proposition 6.1. In the setting fixed for this section, let ebe a tripotent in N. The following are equivalent: eis σ-finite in N; eis σ-finite in M; eis the support tripotent of a normal functional ϕin N*; eis the support tripotent of a τ♯-symmetric normal functional φ in M*.Consequently, for Nσ≔{x∈N:thereexistsaσ-finitetripotenteinNwith{e,e,x}=x}we have Nσ={x∈Mσ:τ(x)=x}=N∩Mσ,and the following are equivalent: Mis σ-finite (that is, Mσ=M); Nis σ-finite (that is, Nσ=N); Ncontains a complete σ-finite tripotent. Proof The implication (b)⇒(a) and the equivalence (c)⇔(d) are clear. The implication (d)⇒(b) follows from [17, Theorem 3.2]. To see (a)⇒(d), let us assume that e is σ-finite in N. Clearly e is the unit in the JBW-algebra N1(e), and since every family of mutually orthogonal projections in this algebra is a family of mutually orthogonal tripotents in N majorized by e, we deduce that e is a σ-finite projection in N1(e). [13, Theorem 4.6] assures the existence of a faithful normal state ϕ in (N1(e))*. By a slight abuse of notation, the symbol ϕ will also denote the functional ϕP1(e). Clearly ϕ∈N* and ϕ∣N1(e) is a faithful normal state. By the arguments above, there exists a τ♯-symmetric normal functional φ in M* such that φ∣N=ϕ. Let e(φ) be the support tripotent of φ in M. We have also commented before this proposition that τ(e(φ))=e(φ) (that is, e(φ)∈N) because ϕ is τ♯-symmetric. Since φ(e)=ϕ(e)=1, we deduce that e≥e(φ). Therefore, e(φ) is a projection in the JBW-algebra N1(e). Furthermore, ϕ(e(φ))=φ(e(φ))=1 and the faithfulness of ϕ∣N1(e) show that e=e(φ). This proves the equivalence of (a), (b), (c) and (d). The equality Nσ=N∩Mσ is clear from the first statement. Since a complete tripotent in N is a complete tripotent in M, the rest of the statement follows from the previous equivalences and [17, Theorem 4.4].□ We can prove now our main result for preduals of real JBW*-triples. Theorem 6.2. The predual of any real JBW*-triple Nis a 1-Plichko space. Moreover, N*is WLD if and only if Nis σ-finite. In the latter case N*is even WCG. Proof We keep the notation fixed for this section with N, M and τ as above. There exists a canonical isometric identification of MR with ((M*)R)*, where any x∈MR acts on (M*)R by the assignment ω↦Reω(x) ( ω∈(M*)R). Thus (M*)R is a real 1-Plichko space and Mσ is again a 1-norming σ-subspace by Theorem 4.1 and [36, Proposition 3.4]. In view of Lemma 2.6 to prove that the predual of N is 1-Plichko, it is enough to show that BN∩Mσ is weak*-dense in BN. Since Mσ is a 1-norming subspace we can easily see that BMσ is weak*-dense in BM. Take an element a∈BN⊂BM. Then there exists a net (aλ)⊂BMσ converging to a in the weak*-topology of M. Since τ is weak*-continuous and Mσ is a norm-closed τ-invariant subspace of M, we can easily see that (aλ+τ(aλ)2)→a in the weak*-topology of M, where (aλ+τ(aλ)2)⊂BNσ=BN∩Mσ, which proves the desired weak*-density. For the last statement, we observe that N is σ-finite if and only if M is (see Proposition 6.1), and hence the desired equivalence follows from Theorem 4.1 and the results presented in Sections 4 and 6. We also note that N σ-finite implies M σ-finite implies M* WCG implies N* WCG, being a complemented subspace.□ We can rediscover the following two results in [4] and [5] as corollaries of our last theorem. Corollary 6.3. ([4, Theorem 1.4]). Let Wbe a von Neumann algebra. Then the predual, (Wsa)*, of the self-adjoint part, Wsa, of Wis a 1-Plichko space. Moreover, (Wsa)*is WLD if and only if Wis σ-finite. In the latter case W*and (Wsa)*are even WCG. Corollary 6.4. ([5, Theorem 1.1]). 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Preduals of JBW*-triples are 1-Plichko spaces

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Abstract We investigate the preduals of JBW*-triples from the point of view of Banach space theory. We show that the algebraic structure of a JBW*-triple M naturally yields a decomposition of its predual M*, by showing that M* is a 1-Plichko space (that is, it admits a countably 1-norming Markushevich basis). In case M is σ-finite, its predual M* is even weakly compactly generated. These results are a common roof for previous results on L1-spaces, preduals of von Neumann algebras, and preduals of JBW*-algebras. 1. Introduction The topic of this paper concerns the interplay between operator algebras, Jordan structures and Banach space theory. The main goal is to prove that the predual of any JBW*-triple satisfies the remarkable Banach space feature called 1-Plichko property (see Theorem 1.1 below). The predual of a JBW*-triple can be viewed as a non-commutative and non-associative generalization of an L1 space. In general such a space may be highly non-separable. Despite this fact, our result implies that the predual of a JBW*-triple admits a nice decomposition into separable subspaces and admits an appropriate Markushevich basis. More precisely, let X be a Banach space. A subspace D⊂X* is said to be a Σ-subspace of X* if there is a linearly dense set S⊂X such that D={ϕ∈X*:{m∈S:ϕ(m)≠0}iscountable}. The Banach space X is called ( r-)Plichko if X* admits an ( r-)norming Σ-subspace, that is there exists a Σ-subspace D of X* such that ∥x∥≤rsup{∣ϕ(x)∣:ϕ∈D,∥ϕ∥≤1}(x∈X) (compare [34, 37]). We prove that the predual M* of any JBW*-triple M is 1-Plichko by identifying a 1-norming Σ-subspace of M=(M*)*. Moreover, the Σ-subspace we find is a canonical one and has an easy algebraic description (see Section 4) and it is even an inner ideal (see Theorem 5.1). This witnesses a close relationship of the algebraic and Banach space structures of JBW*-triples. The 1-Plichko property of a Banach space X is equivalent to the fact that X has a countably 1-norming Markushevich basis [34, Lemma 4.19]. Another deep result [41, Theorem 27] says that X is a 1-Plichko space if and only if it admits a commutative 1-projectional skeleton. A commutative 1-projectional skeleton is a system (Pλ)λ∈Λ of norm one projections on X, where Λ is an up-directed set, fulfilling the following conditions: PλX is separable for each λ and X=⋃λ∈ΛPλX. PλPμ=Pλ whenever λ≤μ. PλPμ=PμPλ for all λ and μ. If (λn) is an increasing net in Λ, it has a supremum, λ∈Λ, and PλX=⋃nPλnX¯. It easily follows that any 1-Plichko space enjoys the 1-separable complementation property saying that any separable subspace can be enlarged to a 1-complemented separable subspace. This property was established by U. Haagerup for preduals of von Neumann algebras with the help of results from modular theory of von Neumann algebras (see [26, Theorem IX.1]). The category of 1-Plichko spaces involves many classes of spaces studied in Banach space theory. Let us recall that X is weakly Lindelöf determined (WLD), if X* is a Σ-subspace of itself. X is called weakly compactly generated (WCG) if it contains a weakly compact subset whose linear span is dense in X. Obviously, every WLD space is 1-Plichko, and it follows from [1, Proposition 2] that every WCG space is WLD. Plichko and 1-Plichko spaces were formally introduced in [34, Section 4.2]. The notion was motivated by a series of papers where A. N. Plichko studied this property under equivalent reformulations (see [48–51]). Although the term 1-Plichko is the most commonly used name for the spaces defined above, they have been also studied under different names. Namely, the class of those Banach spaces which are 1-Plichko is precisely the class termed V by J. Orihuela [44], which has been also studied by M. Valdivia [56]. It has been proved by the third author of this note in [37] that many important spaces have 1-Plichko property, for example L1 spaces for non-negative σ-finite measures, order-continuous Banach lattices, and C(K)-spaces for abelian compact groups K. Moreover, the paper [37] contains the first result on non-commutative L1 spaces showing that the predual of a semi-finite von Neumann algebra is 1-Plichko. Motivated by the latter, the first three authors of this paper prove in [4] that the predual of any von Neumann algebra is 1-Plichko. Moreover, they showed that the canonical 1-norming Σ-subspace is the two-sided ideal of all elements whose range projection is σ-finite. A generalization to JBW*-algebras appeared to be non-trivial. In [5] the same authors showed that the predual of any JBW*-algebra is 1-Plichko. The proof was quite different from that given in the setting of von Neumann algebras. The proof in the Jordan case was based on constructing a special projection skeleton with the help of the set theoretical tool of elementary submodels. Obviously, the question whether, as in the case of von Neumann algebra preduals [4], the result can be obtained without any use of submodels theory is a gap which is not fulfilled by the results in [5]. In the present paper, we prove a further generalization of the above mentioned results by showing that all JBW*-triple preduals are 1-Plichko spaces. Our main result reads as follows. Theorem 1.1. The predual M*of a JBW*-triple Mis a 1-Plichko space. Moreover, M*is WLD if and only if Mis σ-finite. In this case, M* WCG. The approach in this paper resembles more the one of [4] than the one of [5]. One reason for this has already been mentioned, in the present paper the proofs and arguments do not make use of the set theoretic tool of submodels. Moreover, the theory of JBW*-triples allows to connect the description of the Σ-subspace obtained in [4] and to obtain a similar and satisfactory description for JBW*-triples (and hence also for JBW*-algebras), see Theorem 5.1. The key result for this approach is Proposition 4.3. The relevant notions related to JBW*-triples are gathered in Section 2. Theorem 1.1—in fact a more precise version of Theorem 1.1—follows from Theorems 3.1 and 4.1 proved below. Since the second dual of a JB*-triple is a JBW*-triple (see [10, Corollary 3.3.5]), the next result is a straightforward consequence of Theorem 1.1. Corollary 1.2. The dual space of a JB*-triple is a 1-Plichko space. We recall that a Banach space X has the ( r-)separable complementation property if any separable subspace of X is contained in an ( r-)complemented separable subspace of X (compare [26, page 92]). Since 1-Plichko spaces enjoy the 1-separable complementation property (which follows immediately from the characterization using a projectional skeleton formulated above), we also get the following result. Corollary 1.3. Preduals of JBW*-triples have the 1-separable complementation property. The above corollary is an extension of a result of U. Haagerup, who showed that the same statement holds for von Neumann algebra preduals (with different methods, see [26, Theorem IX.1]). 2. Notation and preliminaries In this section, we recall basic notions and results on JBW*-triples and Plichko spaces. We also include some auxiliary results needed to prove our main results. For unexplained notation from Banach space theory, we refer to [21]. The symbols BX and X* will denote the closed unit ball and the dual of a Banach space X, respectively. 2.1. Elements of JBW*-triples W. Kaup [39] obtains an analytic-algebraic characterization of bounded symmetric domains in terms of the so-called JB*-triples, by showing that every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB*-triple. Thanks to this result, JB*-triples offer a natural bridge to connect the infinite-dimensional holomorphy with functional analysis. We recall that a JB*-triple is a complex Banach space E equipped with a continuous ternary product {.,.,.}, which is symmetric and bilinear in the outer variables and conjugate-linear in the middle one, satisfying the following properties: {x,y,{a,b,c}}={{x,y,a},b,c}−{a,{y,x,b},c}+{a,b,{x,y,c}} for all a,b,c,x,y∈E (Jordan identity), the operator x↦{a,a,x} is a Hermitian operator with non-negative spectrum for each a∈E, ∥{a,a,a}∥=∥a∥3 for a∈E. We recall that an operator T∈B(E) is Hermitian if and only if ∥exp(irT)∥=1 for each r∈R. For a,b∈E we define a (linear) operator L(a,b) on E by L(a,b)(x)={a,b,x}, x∈E, and a conjugate-linear operator Q(a,b) by Q(a,b)(x)={a,x,b}. Given a∈E, the symbol Q(a) will denote the operator on E defined by Q(a)=Q(a,a). Every C*-algebra is a JB*-triple with respect to the triple product given by {x,y,z}=12(xy*z+zy*x). The same triple product equips the space B(H,K), of all bounded linear operators between complex Hilbert spaces H and K, with a structure of JB*-triple. Among the examples involving Jordan algebras, we can say that every JB*-algebra is a JB*-triple under the triple product {x,y,z}=(x◦y*)◦z+(z◦y*)◦x−(x◦z)◦y*. An element e in a JB*-triple E is said to be a tripotent if e={e,e,e}. If E is a von Neumann algebra viewed as a JBW*-triple, then any projection is clearly a tripotent; in fact, an element of a von Neumann algebra is a tripotent if and only if it is a partial isometry. For each tripotent e∈E, the mappings Pi(e):E→E(i=0,1,2) defined by P2(e)=L(e,e)(2L(e,e)−idE),P1(e)=4L(e,e)(idE−L(e,e))andP0(e)=(idE−L(e,e))(idE−2L(e,e)) are contractive linear projections (see [23, Corollary 1.2]), called the Peirce projections associated with e. It is known (cf. [10, p. 32]) that P2(e)=Q(e)2, P1(e)=2(L(e,e)−Q(e)2), and P0(e)=idE−2L(e,e)+Q(e)2. In case E is a von Neumann algebra, e∈E a partial isometry, q=e*e the initial projection and p=ee* the final projection, we get P2(e)x=pxq,P1(e)x=px(1−q)+(1−p)xqandP0(e)x=(1−p)x(1−q). If e is even a symmetric element (that is, e*=e) in the von Neumann algebra then we have p=q. The range of Pi(e) is the eigenspace, Ei(e), of L(e,e) corresponding to the eigenvalue i2, and E=E2(e)⊕E1(e)⊕E0(e) is termed the Peirce decomposition of E relative to e. Clearly, e∈E2(e) and Pk(e)(e)=0 for k=0,1. The following multiplication rules (known as Peirce rules or Peirce arithmetic) are satisfied: {E2(e),E0(e),E}={E0(e),E2(e),E}={0}, (2.1) {Ei(e),Ej(e),Ek(e)}⊆Ei−j+k(e), (2.2) where Ei−j+k(e)={0} whenever i−j+k∉{0,1,2} ([23] or [10, Theorem 1.2.44]). A tripotent e is called complete if E0(e)={0}. The complete tripotents of a JB*-triple E are precisely the complex and the real extreme points of its closed unit ball (cf. [6, Lemma 4.1] and [38, Proposition 3.5] or [10, Theorem 3.2.3]). Therefore every JBW*-triple contains an abundant collection of complete tripotents. If E=E2(e), or equivalently, if {e,e,x}=x for all x∈E, we say that e is unitary. For each tripotent e in a JB*-triple, E, the Peirce-2 subspace E2(e) is a unital JB*-algebra with unit e, product a◦eb≔{a,e,b} and involution a⁎e≔{e,a,e} (cf. [10, Section 1.2 and Remark 3.2.2]). As we noticed above, every JB*-algebra is a JB*-triple with respect to the product {a,b,c}=(a◦b*)◦c+(c◦b*)◦a−(a◦c)◦b*. Kaup’s Banach–Stone theorem (see [39, Proposition 5.5]) assures that a surjective operator between JB*-triples is an isometry if and only if it is a triple isomorphism. Consequently, the triple product on E2(e) is uniquely determined by the expression {a,b,c}=(a◦eb⁎e)◦ec+(c◦eb⁎e)◦ea−(a◦ec)◦b⁎e, (2.3) for every a,b,c∈E2(e). Therefore, unital JB*-algebras are in one-to-one correspondence with JB*-triples admitting a unitary element (see also [9, 4.1.55]). A JBW*-triple is a JB*-triple which is also a dual Banach space. Examples of JBW*-triples include von Neumann algebras and JBW*-algebras. Every JBW*-triple admits a unique isometric predual and its triple product is separately weak*-to- weak*-continuous ([3, 29], [10, Theorem 3.3.9]). Consequently, the Peirce projections associated with a tripotent in a JBW*-triple are weak*-to- weak*-continuous. Therefore, for each tripotent e in a JBW*-triple M, the Peirce subspace M2(e) is a JBW*-algebra. Unlike general JB*-triples, JBW*-triples admit a rather concrete representation which we recall in Section 2.4 below as it is the essential tool for proving our results. Let a,b be elements in a JB*-triple E. Following standard terminology, we shall say that a and b are algebraically orthogonal or simply orthogonal (written a⊥b) if L(a,b)=0. If we consider a C*-algebra A as a JB*-triple, then two elements a,b∈A are orthogonal in the C*-sense (that is, ab*=b*a=0) if and only if they are orthogonal in the triple sense. Orthogonality is a symmetric relation. By Peirce arithmetic it is immediate that all elements in E2(e) are orthogonal to all elements in E0(e), in particular, two tripotents u,v∈E are orthogonal if and only if u∈E0(v) (and, by symmetry, if and only if v∈E0(u)). We refer to [8, Lemma 1] for other useful characterizations of orthogonality and additional details not explained here. The order in the partially ordered set of all tripotents in a JB*-triple E is defined as follows: Given two tripotents e,u∈E, we say that e≤u if u−e is a tripotent which is orthogonal to e. Lemma 2.1. ([10, 23, Corollary 1.7, Proposition 1.2.43]). Let u,ebe two tripotents in a JB*-triple E. The following assertions are equivalent. e≤u. P2(e)(u)=e. {u,e,u}=e. {e,u,e}=e. e is a projection (that is, a self-adjoint idempotent) in the JB*-algebra E2(u). For each norm-one functional φ in the predual M*, of a JBW*-triple M, there exists a unique tripotent e∈M satisfying φ=φP2(e) and φ∣M2(e) is a faithful normal state of the JBW*-algebra M2(e) (see [23, Proposition 2]). This unique tripotent e is called the support tripotent of φ, and will be denoted by e(φ). It is explicitly shown in [23] that ifuisatripotentinMwith1=∥φ∥=φ(u),thenu≥e(φ). (2.4) We recall that a subspace I of a JB*-triple E is called an inner ideal, provided {I,E,I}⊆I (that is, provided {a,b,c}∈I whenever a,c∈I and b∈E, see [16]). Clearly, an inner ideal is a subtriple. Note that if e is a tripotent of a JBW*-triple M, then M2(e) is a weak*-closed inner ideal of M(compare the previous Peirce arithmetic). In a von Neumann algebra W (regarded as JBW*-triple) left and right ideals and sets of the form aWb (with fixed a,b∈W) are inner ideals, whereas weak*-closed inner ideals are of the form pWq with projections p,q∈W [15, Theorem 3.16]. Given an element x in a JB*-triple E, the symbol Ex will denote the norm-closed subtriple of E generated by x, that is, the closed subspace generated by all odd powers x[2n+1], where x[1]=x, x[3]={x,x,x}, and x[2n+1]={x,x,x[2n−1]} ( n≥2) (compare, for example, [42, Section 3.3] or [10, Lemma 1.2.10]). It is known that there exists an isometric triple isomorphism Ψ:Ex→C0(L) satisfying Ψ(x)(t)=t, for all t in L (compare [39, 1.15]), where C0(L) is the abelian C*-algebra of all complex-valued continuous functions on L vanishing at 0, L being a locally compact subset of (0,∥x∥] satisfying that L∪{0} is compact. Thus, for any continuous function f:L∪{0}→C vanishing at 0, it is possible to give the usual meaning in the sense of functional calculus to f(x)∈Ex, via f(x)=Ψ−1(f). For each norm-one element x in a JBW*-triple M, r(x) will denote its range tripotent. We succinctly describe its definition. (More details are given for example in [46, Section 2.2] or in [14, comments before Lemma 3.1] or [8, Section 2]). For x∈M with ∥x∥=1, the functions t→t12n−1 give rise to an increasing sequence (x[12n−1]) which weak*-converges to r(x) in M. The tripotent r(x) is the smallest tripotent e∈M satisfying that x is a positive element in the JBW*-algebra M2(e) (see, for example, [14, comments before Lemma 3.1] or [8, Section 2]). The inequality x≤r(x) holds in M2(r(x)) for every norm-one element x∈E. For a non-zero element z∈M, the range tripotent of z, r(z), is precisely the range tripotent of z∥z∥, and we set r(0)=0. Let M be a JBW*-triple. We recall that a tripotent u in M is said to be σ-finite if u does not majorize an uncountable orthogonal subset of tripotents in M. Equivalently, u is a σ-finite tripotent in M if and only if there exists an element φ in M* whose support tripotent e(φ) coincides with u (cf. [17, Theorem 3.2]). Following standard notation, we shall say that M is σ-finite if every tripotent in M is σ-finite, equivalently, every orthogonal subset of tripotents in M is countable (cf. [17, Proposition 3.1]). It is also known that the sum of an orthogonal countable family of mutually orthogonal σ-finite tripotents in M is again a σ-finite tripotent (see [17, Theorem 3.4(i)]). It is further proved in [17, Theorem 3.4(ii)] that every tripotent in M is the supremum of a set of mutually orthogonal σ-finite tripotents in M. When a von Neumann algebra W is regarded as a JBW*-triple, a projection p is σ-finite in the triple sense if and only if it is σ-finite or countably decomposable in the usual sense employed for von Neumann algebras (compare [53, Definition 2.1.8] or [55, Definition II.3.18]). We will need the following properties of σ-finite tripotents which have been borrowed from [17]. Lemma 2.2. ([17]). Let Mbe a JBW*-triple and let ebe a tripotent of M. Then the following hold: M2(e)is a JBW*-subtriple of Mand any tripotent p∈M2(e)is σ-finite in M2(e)if and only if it is σ-finite in M. eis σ-finite if and only if M2(e)is σ-finite. If eis σ-finite, then any tripotent in M2(e)is σ-finite in M. Proof Since M2(e) is a weak*-closed subtriple of M, assertion (i) follows from [17, Lemma 3.6(ii)]. Assertion (ii) follows from (i), [17, Theorem 4.4(viii)–(ix)] and the fact that e is a complete tripotent in M2(e). Finally, assertion (iii) follows immediately from (i) and (ii).□ For non-explained notions concerning JB*-algebras and JB*-triples, we refer to the monographs [9, 10]. 2.2. Contractive and bicontractive projections One of the main properties enjoyed by any member E in the class of JB*-triples states that the image of a contractive projection P:E→E (where contractive means ∥P∥≤1) is again a JB*-triple with triple product {x,y,z}P≔P({x,y,z}) for x,y,z in P(E) and P{a,x,b}=P{a,P(x),b},a,b∈P(E),x∈E (2.5) (see [24, 35, 49]). It is further known that under these conditions P(E) need not be, in general, a JB*-subtriple of E (compare [22, Example 1] or [40, Example 3]). But note that if P(E) is known to be a subtriple, then {·,·,·}P coincides with the original triple product of E because in JB*-triples norm and triple product determine each other (see e.g. [10, Theorem 3.1.7, 3.1.20]). Fortunately, more can be said about the JB*-triple structure of P(E). It is known that P(E) is isometrically isomorphic to a JB*-subtriple of E** (see [25, Theorem 2]). If P:E→E is even a bicontractive projection (where bicontractive means ∥P∥≤1 and ∥I−P∥≤1—by IV or simply I we denote the identity on a vector space V) on a JB*-triple, it satisfies a stronger property. Namely, P(E) is then a JB*-subtriple of E, in particular (2.5) can be improved because the identities P{a,b,x}={a,b,P(x)}andP{a,x,b}={a,P(x),b} (2.6) hold for a,b∈P(E), x∈E (cf. [25, Section 3]). It is further known that when P is bicontractive, there exists a surjective linear isometry (that is, a triple automorphism) Θ on E of period 2 such that P=12(I+Θ) (see [25, Theorem 4]). Since, by another interesting property of JBW*-triples, every surjective linear isometry on a JBW*-triple is weak*-to- weak*-continuous (see [29, Proof of Theorem 3.2]) we have, as a consequence, that a bicontractive projection P on a JBW*-triple is weak*-to- weak*-continuous. 2.3. Von Neumann tensor products We recall now some basic facts on von Neumann tensor products of von Neumann algebras. The theory has been essentially borrowed from [55, Chapter IV], and we refer to the latter monograph for additional results not commented here. Let A⊂B(H) and W⊂B(K) be von Neumann algebras. The algebraic tensor product A⊗W is canonically embedded into B(H⊗K), where H⊗K is the Hilbertian tensor product of H and K (see [55, Definition IV.1.2]). The von Neumann algebra generated by the algebraic tensor product A⊗W is denoted A⊗¯W, and is called the von Neumann tensor product of A and W. Note that A⊗¯W is the weak* closure of A⊗W in B(H⊗K) (see [55, Section IV.5]). If A is commutative, then the predual of A⊗¯W is canonically identified with the projective tensor product of preduals, that is (A⊗¯W)*=A*⊗^πW*. (2.7) This follows from [55, Theorem IV.7.17] (or rather [55, Section IV.7]). Furthermore, the special case of a separable W* is treated in [53, Theorem 1.22.13], while there is another approach via results on operator spaces: Results due to E. G. Effros and Z. J. Ruan show that equality (2.7) holds for any von Neumann algebra W, when the projective tensor product on the right-hand side is in the category of operator spaces ([18, Theorem 7.2.4], [19]). But if A is commutative, it carries the minimal operator-space structure by [18, Proposition 3.3.1], and hence the predual A* carries the maximal structure by [18, (3.3.13) or (3.3.15) on p. 51], and hence by [18, (8.2.4) on p. 146] the projective tensor product in the category of operator spaces coincides with the projective tensor product in the Banach space sense. Lemma 2.3. Let Aand Wbe von Neumann algebras with Acommutative. Suppose P:W→Wis a weak*-to- weak*-continuous contractive projection. Then the following holds: P(W)is a JBW*-triple with triple product {x,y,z}P≔P({x,y,z})for x,y,zin P(W). A⊗¯P(W), the weak*-closure of the algebraic tensor product A⊗P(W)in A⊗¯W, is the range of a weak*-to- weak*-continuous contractive projection Q on A⊗¯W. A⊗¯P(W)is a JBW*-triple with triple product {x,y,z}Q≔Q({x,y,z})for x,y,zin A⊗¯P(W). Moreover, (A⊗¯P(W))*=A*⊗^π(P(W))*=A*⊗^πP*(W*). Proof We know from Section 2.2 that statement (i) is satisfied. Since P is weak*-to- weak* continuous, it is the dual map of a map P*:W*→W*. It is clear that P* is a contractive projection on W*. It follows from basic tensor product properties (cf. [11, 3.2] or [52, Proposition 2.3]) that I⊗P* is a contractive projection on A*⊗^πW*. Moreover, by [11, 3.8] or [52, Proposition 2.5] the norm on its range (which is the norm-closure of the algebraic tensor product A*⊗P*(W*)) is the projective norm coming from A*⊗^πP*(W*). Further, it is clear that the dual mapping Q=(I⊗P*)* is a weak*-to- weak*-continuous contractive projection on (A*⊗^πW*)*=A⊗¯W. Using the results commented in Section 2.2 we know that its range is a JBW*-triple with the triple product defined in (iii). Since the range of Q is canonically identified with the dual of A*⊗^πP*(W*), to complete the proof of (ii) and (iii) it is enough to show that the range of (I⊗P*)* is A⊗¯P(W). To show the desired equality we observe that the restriction of (I⊗P*)* to the algebraic tensor product A⊗W coincides with I⊗P. Therefore the range of (I⊗P*)* contains A⊗P(W) and hence also its weak* closure A⊗¯P(W). Conversely, since the unit ball BA⊗W is weak*-dense in BA⊗¯W (for example by the Kaplansky density theorem), and (I⊗P*)* is weak*-to- weak*-continuous, BA⊗W is weak* dense in the unit ball of the range of (I⊗P*)* as well. This completes the proof.□ Lemma 2.4. Let Aand Wbe von Neumann algebras with Acommutative. Suppose P:W→Wis a bicontractive projection. Then the following holds: P(W)is a JBW*-subtriple of W. A⊗¯P(W), the weak*-closure of the algebraic tensor product A⊗P(W)in A⊗¯W, is the range of a bicontractive projection on A⊗¯W. A⊗¯P(W)is a JBW*-subtriple of A⊗¯Wand, moreover, (A⊗¯P(W))*=A*⊗^π(P(W))*=A*⊗^πP*(W*). Proof By Section 2.2, we know that P(W) is a JB*-subtriple of W and that P is weak*-to- weak*-continuous. Hence we can apply Lemma 2.3. Moreover, since P is even bicontractive, we get that P* is bicontractive, and hence I⊗P* and Q=(I⊗P*)* are bicontractive too. Finally, since Q is bicontractive, by Section 2.2 we get that A⊗¯P(W) is a JBW*-subtriple of A⊗¯W.□ 2.4. Structure theory In this subsection, we recall an important structure result, due to G. Horn [30] and G. Horn and E. Neher [31], which allows us to represent every JBW*-triple in a concrete way. These results will be the main tool for proving that JBW*-triple preduals are 1-Plichko spaces. We begin by recalling the definition of Cartan factors. There are six types of them (compare [10, Example 2.5.31]): Type 1: A Cartan factor of type 1 coincides with a Banach space B(H,K), of all bounded linear operators between two complex Hilbert spaces H and K, where the triple product is defined by {x,y,z}=2−1(xy*z+zy*x). If dimH=dimK, then we can suppose H=K and we get the von-Neumann algebra B(H). If dimK<dimH, we may suppose that K is a closed subspace of H and then B(H,K) is a JB*-subtriple of B(H). Moreover, if p is the orthogonal projection of H onto K, then x↦px is a bicontractive projection of B(H) onto B(H,K). If dimK>dimH, we may suppose that H is a closed subspace of K, p the orthogonal projection of K onto H and then x↦xp is a bicontractive projection of B(K) onto B(H,K). Types 2 and 3: Cartan factors of types 2 and 3 are the subtriples of B(H) defined by C2={x∈B(H):x=−jx*j} and C3={x∈B(H):x=jx*j}, respectively, where j is a conjugation (that is, a conjugate-linear isometry of period 2) on H. If j is a conjugation on H, then there is an orthonormal basis (eγ)γ∈Γ such that j(∑γ∈Γcγeγ)=∑γ∈Γcγ¯eγ. Each x∈B(H) can be represented by a ‘matrix’ (xγδ)γ,δ∈Γ. It is easy to check that the representing matrix of jx*j is the transpose of the representing matrix of x. Hence, C2 consists of operators with antisymmetric representing matrix and C3 of operators with symmetric ones. Therefore, P(x)=12(xt+x) (where xt=jx*j is the transpose of x with respect to the basis chosen above) is a bicontractive projection on B(H) such that C3 is the range of P, and C2 is the range of I−P. Type 4: A Cartan factor of type 4 (denoted by C4) is a complex spin factor, that is, a complex Hilbert space (with inner product ⟨.,.⟩) provided with a conjugation x↦x¯, triple product {x,y,z}=⟨x,y⟩z+⟨z,y⟩x−⟨x,z¯⟩y¯, and norm given by ∥x∥2=⟨x,x⟩+⟨x,x⟩2−∣⟨x,x¯⟩∣2. We point out that C4 is isomorphic to a Hilbert space and hence, in particular, reflexive. Types 5 and 6: All we need to know about Cartan factors of types 5 and 6 (also called exceptional Cartan factors) is that they are all finite dimensional. Although H. Hanche-Olsen showed in [27, Section 5] that the standard method to define tensor products of JC-algebras (and JW *-triples) is, in general, hopeless, von Neumann tensor products can be applied in the representation theory of JBW *-triples. Let A be a commutative von Neumann algebra and let C be a Cartan factor which can be realized as a JW*-subtriple of some B(H). As before, the symbol A⊗¯C will denote the weak*-closure of the algebraic tensor product A⊗C in the usual von Neumann tensor product A⊗¯B(H) of A and B(H). This applies to Cartan factors of types 1–4 (this is obvious for Cartan factors of types 1–3, the case of type 4 Cartan factors follows from [28, Theorem 6.2.3]). The above construction does not cover Cartan factors of types 5 and 6. When C is an exceptional Cartan factor, A⊗¯C will denote the injective tensor product of A and C, which can be identified with the space C(Ω,C), of all continuous functions on Ω with values in C endowed with the pointwise operations and the supremum norm, where Ω denotes the spectrum of A (cf. [52, p. 49]). We observe that if C is a finite dimensional Cartan factor which can be realized as a JW*-subtriple of some B(H) both definitions above give the same object (cf. [55, Theorem IV.4.14]). The structure theory settled by G. Horn and E. Neher [30, 31, (1.7)] proves that every JBW*-triple M writes (uniquely up to triple isomorphisms) in the form M=(⨁j∈JAj⊗¯Cj)ℓ∞⊕ℓ∞H(W,α)⊕ℓ∞pV, (2.8) where each Aj is a commutative von Neumann algebra, each Cj is a Cartan factor, W and V are continuous von Neumann algebras, p is a projection in V, α is a linear involution on W commuting with *, that is, a linear *-antiautomorphism of period 2 on W, and H(W,α)={x∈W:α(x)=x}. 2.5. Some facts on Plichko spaces The following lemma sums up several basic properties of Σ-subspaces. Lemma 2.5. Let Xbe a Banach space and D⊂X*a Σ-subspace. Then the following hold: Dis weak*-countably closed. That is, C¯w*⊂Dwhenever C⊂Dis countable. In particular, Dis weak*-sequentially closed and norm-closed. Bounded subsets of Dare weak*-Fréchet-Urysohn. That is, given A⊂Dbounded and x*∈Dsuch that x*∈A¯w*, then there is a sequence (xn*)in A weak*-converging to x*. Let D′⊂X*be any other subspace satisfying (i) and (ii). If D∩D′is 1-norming, then D=D′. If Xis WLD, then X*is the only norming Σ-subspace of X*. If Dis 1-norming, then for any x∈Xthere is x*∈Dof norm one such that x*(x)=∥x∥. Proof Assertion (i) follows from the very definition of a Σ-subspace, assertion (ii) follows from [34, Lemma 1.6]. Assertion (iii) is an easy consequence of (i) and (ii) and follows from [35, Lemma 2] (in fact in the just quoted lemma it is assumed that D′ is a Σ-subspace as well, but the proof uses only properties (i) and (ii)). Assertion (iv) follows immediately from (iii) and (v) is an easy consequence of (i).□ We will also need the following easy lemma on quotients of 1-Plichko spaces. Lemma 2.6. Let Xbe a 1-Plichko Banach space, and let D⊂X*be a 1-norming Σ-subspace. Suppose that Z⊂X*is a weak*-closed subspace such that D∩BZis weak*dense in BZ. Then D∩Zis a 1-norming Σ-subspace of Z=(X/Z⊥)*. Proof Since Z is a weak*-closed subspace of the dual space X*, it is canonically isometrically identified with (X/Z⊥)*. Further, by the assumptions it is clear that D∩Z is a 1-norming subspace of Z. It remains to show it is a Σ-subspace. To do that, fix a linearly dense set S⊂X such that D={x*∈X*:{x∈S:x*(x)≠0}iscountable}. Let S˜ be the image of S in X/Z⊥ by the canonical quotient mapping. It is clear that S˜ is linearly dense. Let D˜={x*∈Z=(X/Z⊥)*:{x∈S˜:x*(x)≠0}iscountable} be the Σ-subspace induced by S˜. It is easy to check that D∩Z⊂D˜. It follows from Lemma 2.5(iii) that D∩Z=D˜, which completes the proof.□ 3. Preduals of σ-finite JBW*-triples The aim of this section is to prove the following result. Theorem 3.1. The predual of any σ-finite JBW*-triple is WCG, in fact even Hilbert-generated. Recall that a Banach space X is said to be Hilbert-generated if there is a Hilbert space H and a bounded linear mapping T:H→X with dense range. It is clear that any Hilbert-generated Banach space is WCG (the generating weakly compact set is precisely T(BH)). Theorem 3.1 above follows from the following stronger statement, which is a JBW*-triple analog of [4, Lemma 3.3] for von Neumann algebras and of [5, Proposition 3.7] for JBW*-algebras. Proposition 3.2. Let ebe a σ-finite tripotent in a JBW*-triple M. Then the predual of the space M2(e)⊕M1(e) (i.e. (P2(e)+P1(e))*(M*)) is Hilbert-generated. To see that Theorem 3.1 follows from the above proposition it is enough to use the fact that any JBW*-triple contains an abundant set of complete tripotents. In particular, any σ-finite JBW*-triple M contains a σ-finite complete tripotent e∈M such that M=M2(e)⊕M1(e). Hence Proposition 3.2 entails Theorem 3.1. Next let us focus on the proof of Proposition 3.2. Similarly as in the case of von Neumann algebras and JBW*-algebras, it will be done by introducing a canonical (semi)definite inner product. Barton and Friedman [2, Proposition 1.2] showed that given an element φ in the dual of a JB*-triple E and an element z∈E such that φ(z)=∥φ∥=∥z∥=1, the map E×E∋(x,y)↦⟨x,y⟩φ≔φ{x,y,z} defines a hermitian semi-positive sesquilinear form with the associated pre-hilbertian seminorm ∥x∥φ≔(φ{x,x,z})1/2 on M and is independent of z. We shall need the following technical lemma borrowed from [17, Lemma 4.1]: Lemma 3.3. Let Mbe a JBW*-triple, let φ∈M*be of norm one and let e=e(φ)∈Mbe its support tripotent. Then the annihilator of the pre-Hilbertian seminorm ∥·∥φis precisely M0(e), that is, {x∈M:∥x∥φ=0}=M0(e). (3.1)In particular, the restriction of ∥·∥φto M2(e)⊕M1(e)is a pre-Hilbertian norm and the restriction of ⟨·,·⟩φto M2(e)⊕M1(e)is an inner product. Proof The first statement is proved in [17, Lemma 4.1], the positive definiteness of ∥·∥φ and of ⟨·,·⟩φ on M2(e)⊕M1(e) follows immediately (see also [23, Lemma 1.5], [45]).□ Now we are ready to prove the main proposition of this section: Proof of Proposition 3.2 Since e is a σ-finite tripotent, there exists a norm-one normal functional φ∈M* such that e=e(φ) is the support tripotent of φ. Denote by hφ the pre-Hilbertian space M2(e)⊕M1(e) equipped with the inner product ⟨·,·⟩φ=φ{·,·,e}, and write Hφ for its completion. Let us first consider Φ˜(a) defined by x↦⟨x,a⟩φ for a∈hφ, x∈M. By the Cauchy-Schwarz inequality we have ∣Φ˜(a)(x)∣=∣⟨x,a⟩φ∣≤∥x∥φ∥a∥φ≤∥x∥∥a∥φ which, together with the separate w*-continuity of the triple product, shows that Φ˜ is a well-defined conjugate-linear contractive map from hφ to M*. In order to see that the range of Φ˜ is contained in (M2(e)⊕M1(e))*=(P2*(e)+P1*(e))(M*), let us observe that for any a∈hφ and y∈M0(e), we have ∥y∥φ=0 by Lemma 3.3, and hence Φ˜(a)(y)=0. Thus, by density of hφ in Hφ, Φ˜=(P2*(e)+P1*(e))Φ˜ gives rise to a conjugate-linear continuous map Φ:Hφ→(M2(e)⊕M1(e))*. We shall finally prove that Φ has norm-dense range. Suppose z∈M2(e)⊕M1(e) satisfies Φ(a)(z)=0 for every a∈hφ. In particular, 0=Φ(z)(z)=∥z∥φ2 and thus, by Lemma 3.3, z=0. By the Hahn-Banach theorem, Φ has dense range. If we replace the map Φ by Φj, where j is a conjugation on Hφ, then we have a linear mapping.□ 4. The case of general JBW*-triples In this section, we state and prove Theorem 4.1, which gives a more precise version of the first part of Theorem 1.1. To provide a precise formulation, we introduce one more notation. For a JBW* triple M we define the set Mσ={x∈M:thereisaσ-finitetripotente∈MsuchthatP2(e)x=x} and note that Mσ={x∈M:thereisaσ-finitetripotente∈Msuchthat{e,e,x}=x}={x∈M:r(x)isaσ-finitetripotent}. Indeed, the first equality follows from the fact that the range of P2(e) is the eigenspace of L(e,e) corresponding to the eigenvalue 1. Let us show the second equality. The inclusion ‘ ⊃’ is obvious. To show the converse inclusion, let x∈Mσ. Fix a σ-finite tripotent e∈M with x=P2(e)x, that is, x∈M2(e). Since M2(e) is a JBW*-subtriple of M and r(x) belongs to the JBW*-subtriple generated by x, we have r(x)∈M2(e) and so r(x) is σ-finite by Lemma 2.2. We mention the easy but useful fact that Mσ is 1-norming in M=(M*)*. To see this we simply observe that Mσ contains all σ-finite tripotents of M, or equivalently, all support tripotents of functionals in M*. Theorem 4.1. The predual space of a JBW*-triple Mis a 1-Plichko space. Moreover, Mσisa1-normingΣ-subspaceofM=(M*)*. (4.1)In particular, M*is WLD if and only if Mis σ-finite. It is not obvious that Mσ is a subspace, but this will follow by the proof of Theorem 4.1; it will be reproved a second time in Theorem 5.1. The ‘in particular’ part of the theorem is an immediate consequence of the first statements of the theorem. Indeed, M is σ-finite if and only if M=Mσ (cf. Lemma 2.2). Hence, if M is σ-finite, then M* is WLD by the first statement. Conversely, if M* is WLD, then by the first part of the theorem together with Lemma 2.5 (iv) we get M=Mσ, hence M is σ-finite. Thus, it is enough to prove (4.1). This will be done in the rest of this section by using results in [4] and the decomposition (2.8). The following proposition is almost immediate from the main results of [4]. Proposition 4.2. The statement of Theorem4.1holds for von Neumann algebras. Proof It is enough to show (4.1) in case M is a von Neumann algebra. In view of [4, Proposition 4.1], to this end it is enough to show that Mσ={x∈M:x=qxqforaσ-finiteprojectionq∈M}. Let x be in the set on the right-hand side. Fix a σ-finite projection q∈M with x=qxq. Then q is a σ-finite tripotent and {q,q,x}=12(qx+xq)=qxq=x. Hence x∈Mσ. Conversely, let x∈Mσ and let u∈M be a σ-finite triponent with x=P2(u)x. Since M is a von Neumann algebra, u is a partial isometry and hence P2(u)x=pxq, where p=uu* is the final projection and q=u*u is the initial projection. Then p is a σ-finite projection. Indeed, suppose that (rγ)γ∈Γ is an uncountable family of pairwise orthogonal projections smaller than p. Then it is easy to check that (rγu)γ∈Γ is an uncountable family of pairwise orthogonal tripotents smaller than u. Similarly we get that q is σ-finite. Hence their supremum r=p∨q is σ-finite as well ([17, Theorem 3.4] or [33, Exercice 5.7.45]) and satisfies x=rxr. Thus x belongs to the set on the right-hand side and the proof is complete.□ Proposition 4.3. Let P:M→Mbe a bicontractive projection on a JBW*-triple, let N=P(M), and let ebe a tripotent in N. Then eis σ-finite in Nif and only if eis σ-finite in M, that is, Nσ=N∩Mσ. Proof The ‘if’ implication is clear. Let e be a σ-finite tripotent in N. By [17, Theorem 3.2] there exists a norm-one functional ϕ∈N* whose support tripotent in N is e. Let us define ψ=P*(ϕ)=ϕP∈M*. Clearly ∥ψ∥=1. We shall prove that e is the support tripotent of ψ in M, and hence e is σ-finite in M ([17, Theorem 3.2]). Let u be the support tripotent of ψ in M. From ψ(e)=ϕ(e)=1=∥ψ∥ we get e≥u (compare [23, part (b) in the proof of Proposition 2]). We set u1=P(u) and u2=u−u1. Since e≥u in M, we deduce that {e,u,e}=u={e,e,u} ( e−u∈M0(u) and Peirce rules). Hence, u1=P(u)={e,Pu,e}={e,u1,e} and u1={e,e,u1} by (2.6). It follows that u1={e,u1,e}∈M2(e) and that u1={e,u1,e}=u1⁎e is a hermitian element in the closed unit ball of the JBW*-algebra N2(e). As e is the unit in this algebra and u1 is a hermitian element of norm less than one, we see that e−u1 is a positive element in the JBW*-algebra N2(e). The condition ϕ(e)=1=ψ(u)=ϕP(u)=ϕ(u1) implies, by the faithfulness of ϕ∣N2(e), that u1=e. It follows from the above that u2={e,e,u}−{e,e,u1}={e,e,u2} and similarly u2={e,u2,e}. These identities combined with the fact that u=e+u2 is a tripotent (that is, {e+u2,e+u2,e+u2}=e+u2) yield e+u2=e+2{u2,u2,e}+{u2,e,u2}+3u2+{u2,u2,u2}. After applying the bicontractive projection I−P in both terms of the last equality we get −2u2={u2,u2,u2}. Now 2∥u2∥=∥{u2,u2,u2}∥=∥u2∥3 implies either u2=0 or ∥u2∥2=2. The latter is not possible because ∥u2∥≤1 by the fact that u2=(I−P)u and I−P is a contraction. Thus u2=0, and hence e=u, which proves the first statement. For the last identity, we observe that for every element x∈N, its range tripotent r(x) (in N or in M) lies in N. Suppose x is an element in N whose range tripotent is σ-finite in N. We deduce from the first statement that r(x) is also σ-finite in M, and hence Nσ⊆Mσ. The inclusion Nσ⊇Mσ∩N is clear.□ By combining Proposition 4.2, Proposition 4.3 and Lemma 2.6 we get the following proposition. Proposition 4.4. Let P:W→Wbe a bicontractive projection on a von Neumann algebra W, let M=P(W). Then M*is a 1-Plichko space. Furthermore, Mσis a 1-norming Σ-subspace of M. Now we are ready to prove the validity of (4.1) for most of the summands from the representation (2.8): Proposition 4.5. Let Mbe a JBW*-triple of one of the following forms: M=A⊗¯C, where Ais a commutative von Neumann algebra and Cis a Cartan factor of type 1, 2 or3. M=H(W,α), where Wis a von Neumann algebra and αis a linear involution on Wcommuting with *. M=pV, where Vis a von Neumann algebra and p∈Vis a projection. Then Mσ is a 1-norming Σ-subspace of M=(M*)*. Proof We will apply Proposition 4.4. To do that it is enough to show that M is the range of a bicontractive projection on a von Neumann algebra. (a) If C is a Cartan factor of type 1, 2 or 3, then C is the range of a bicontractive projection on a certain von Neumann algebra W, as it was previously observed after the definitions of the respective Cartan factors. The desired bicontractive projection on A⊗¯W is finally given by Lemma 2.4. (b) A bicontractive projection on W is given by x↦12(x+α(x)). (c) The mapping x↦px defines a bicontractive projection on V.□ The remaining summands from (2.8) are covered by the following theorem, which we formulate in a more abstract setting of Banach spaces. Theorem 4.6. Let (Ω,Σ,μ)be a measure space with a non-negative semifinite measure, and let Ebe a reflexive Banach space. Then the space L1(μ,E)of Bochner-integrable functions is 1-Plichko. Furthermore, L1(μ,E)is WLD if and only if μis σ-finite, in the latter case it is even WCG. More precisely, there is a family of finite measures (μγ)γ∈Γsuch that L1(μ,E)is isometric to (⨁γ∈ΓL1(μγ,E))ℓ1and D={f=(fγ)γ∈Γ∈(⨁γ∈ΓL∞(μγ,E))ℓ∞:{γ∈Γ:fγ≠0}iscountable}is a 1-norming Σ-subspace of (L1(μ,E))*=(⨁γ∈ΓL∞(μγ,E))ℓ∞. Proposition 4.7. Let μbe a finite measure, and let Ebe a reflexive Banach space. Then L1(μ,E)is WCG. Proof The proof is done similarly as in the scalar case (cf. [37, Theorem 5.1]). Let us consider the identity mapping T:L2(μ,E)→L1(μ,E). By the Cauchy-Schwarz inequality we get ∥T∥≤∥μ∥, hence T is a bounded linear operator. Moreover, the range of T is dense, since countably valued functions in L1(μ,E) are dense in the latter space. Finally, L2(μ,E) is reflexive because E and E* have Radon-Nikodým property (see [12, Theorem IV.1.1]). Thus, L1(μ,E) is indeed WCG.□Remark: Note that if E is isomorphic to a Hilbert space, then we can even conclude that L1(μ,E) is Hilbert-generated, since in this case L2(μ,E) is also isomorphic to a Hilbert space. Indeed, if E is even isometric to a Hilbert space, the norm on L2(μ,E) is induced by the scalar product ⟨f,g⟩=∫⟨f(ω),g(ω)⟩dμ(ω). Proof of Theorem 4.6 We imitate the proof of [37, Theorem 5.1]. Let B⊂Σ be a maximal family with the following properties: 0<μ(B)<+∞ for each B∈B; μ(B1∩B2)=0 for each B1,B2∈B distinct. The existence of such a family follows immediately from Zorn’s lemma. Take any separable-valued Σ-measurable function f:Ω→E. Then clearly ∫∥f(ω)∥dμ(ω)=∑B∈B∫B∥f(ω)∥dμ(ω). Therefore, L1(μ,E) is isometric to the ℓ1-sum of spaces L1(μ∣B,E), B∈B. Since μ∣B is finite for each B∈B, L1(μ∣B,E) is WCG (and hence WLD) by the previous Proposition 4.7. Further, it is clear that the dual of L1(μ,E) is canonically isometric to the ℓ∞-sum of the family {(L1(μ∣B,E))*:B∈B}. More concretely, since E is reflexive, by [12, Theorem IV.1.1] we have (L1(μ∣B,E))*=L∞(μ∣B,E*) for each B∈B, and hence L1(μ,E)*=(⨁B∈BL∞(μ∣B,E*))ℓ∞. Finally, it follows from [34, Lemma 4.34] that D={(fB)B∈B∈(⨁B∈BL∞(μ∣B,E*))ℓ∞:{B∈B;fB≠0}iscountable} is a 1-norming Σ-subspace of (L1(μ,E))*. To prove the last statement, it is enough to observe that μ is σ-finite if and only if B is countable, that a countable ℓ1-sum of WCG spaces is again WCG and that an uncountable ℓ1-sum of nontrivial spaces contains ℓ1(ω1) and hence is not WLD. (Recall that WLD property passes to subspaces.)□ Proposition 4.8. Let Abe a commutative von Neumann algebra and Ca Cartan factor. Then (A⊗¯C)*=A*⊗^πC*. Proof If C is a Cartan factor of type 1, 2 or 3, then C is the range of a bicontractive projection on a von Neumann algebra and hence the equality follows from Lemma 2.4. If C is a type 4 Cartan factor, it follows from [20, Lemma 2.3] that C is the range of a (unital positive) contractive projection P:B(H)→B(H) where H is an appropriate Hilbert space. The mapping P**:B(H)**→B(H)** is a weak*-to- weak*-continuous contractive projection on the von Neumann algebra B(H)** whose range is C by (Goldstine’s theorem and) reflexivity of C. Hence the desired equality follows from Lemma 2.3. If C is a Cartan factor of type 5 or 6, then it is finite dimensional and A⊗¯C is defined to be the injective tensor product. Further, by [11, 3.2] or [52, p. 24] we get (A*⊗^πC*)*=B(A*,C) which coincides with the injective tensor product A⊗^εC, as C has finite dimension.□ Lemma 4.9. Let (Mγ)γ∈Γbe an indexed family of JBW*-triples, and let us denote M=(⨁γ∈ΓMγ)ℓ∞. Then Mσ={(xγ)γ∈Γ∈M:xγ∈(Mγ)σforγ∈Γand{γ∈Γ:xγ≠0}iscountable}. Proof This follows easily if we observe that e=(eγ)γ∈Γ∈M is a tripotent if and only if eγ is a tripotent for each γ and, moreover, e is σ-finite if and only if each eγ is σ-finite and only countably many eγ are non-zero.□ Proposition 4.10. Let Abe a commutative von Neumann algebra and Ca reflexive Cartan factor. (This applies, in particular, to Cartan factors of types 4, 5 and 6.) Let M=A⊗¯C. Then Mσis a 1-norming Σ-subspace of M=(M*)*, and hence M*is 1-Plichko. Furthemore, M*is WLD if and only if Ais σ-finite. In such a case M*is even WCG. Proof If A is a commutative von Neumann algebra, by [55, Theorem III.1.18] it can be represented as L∞(Ω,μ), where Ω is a locally compact space and μ a positive Radon measure on Ω. In fact, Ω is the topological sum of a family of compact spaces (Kγ)γ∈Γ. Then the predual of A is identified with L1(Ω,μ)=(⨁γ∈ΓL1(Kγ,μ∣Kγ))ℓ1. Since (A⊗¯C)*=A*⊗^πC*=L1(μ,C*), we can use Theorem 4.6. To complete the proof it is enough to show that D=Mσ, where D is the Σ-subspace provided by Theorem 4.6. Since M=(⨁γ∈ΓL∞(Kγ,μ∣Kγ,C))ℓ∞, due to Lemma 4.9, it is enough to show that L∞(μ,C) is σ-finite whenever μ is finite. But, in this case, its predual, L1(μ,C*), is WCG by Proposition 4.7, thus L∞(μ,C) is σ-finite by Theorem 4.6.□ Proof of Theorem 4.1 We have already mentioned that it is enough to show (4.1). Let M be a JBW*-triple and consider the decomposition (2.8). By Propositions 4.5 and 4.10 each summand fulfills (4.1). Further, Lemma 4.9 and [34, Lemma 4.34] yield the validity of (4.1) for M.□ In passing we remark that from Theorem 4.1 (and the general facts on Plichko spaces), we have that Mσ is norm-closed and even weak*-countably closed; it is additionally weak*-closed if and only if M is σ-finite. 5. Structure of the space Mσ In the previous section we proved that, for any JBW*-triple M, Mσ is a 1-norming Σ-subspace of M=(M*)*. If M is σ-finite, it is the only 1-norming Σ-subspace and coincides with the whole M. If M is not σ-finite, there may be plenty of different 1-norming Σ-subspaces (cf. [34, Example 6.9]). However, Mσ is the only canonical 1-norming Σ-subspace. What we mean by this statement is in the content of the following theorem. Theorem 5.1. Let Mbe a JBW*-triple. Then Mσis a norm-closed inner ideal in M. Moreover, it is the only 1-norming Σ-subspace which is also an inner ideal. The theorem will be proved at the end of this section. The following technical result provides a characterization of σ-finite tripotents which is required later. We recall that, given a tripotent u in a JBW*-triple M, there exists a complete tripotent w∈M such that u≤w (see [29, Lemma 3.12(1)]). Proposition 5.2. Let ube a tripotent in a JBW*-triple M. The following statements are equivalent: uis σ-finite; There exist a σ-finite tripotent vand a complete tripotent win Msuch that v≤wand (w−v)⊥u. Proof The implication (a)⇒(b) is clear with v=u and any complete tripotent w in M with u≤w . (b)⇒(a) Suppose there exist a σ-finite tripotent v and a complete tripotent w in M such that v≤w and (w−v)⊥u. Writing w=v+(w−v) and using successively the orthogonality of w−v to u and to v we obtain {w,w,u}={w,v,u}={v,v,u}, and hence L(w,w)u=L(v,v)u, and similarly {w,u,w}={v,u,v}. Since w−v⊥M2(v)∋{v,u,v}, it follows that P2(w)(u)=Q(w)2(u)={w,{v,u,v},w}={v,{v,u,v},v}=P2(v)(u).Therefore, P2(w)(u)=P2(v)(u) and P1(w)(u)=2L(w,w)(u)−2P2(w)(u)=P1(v)(u). The completeness of w assures that u=P2(w)(u)+P1(w)(u)=P2(v)(u)+P1(v)(u) lies in M2(v)⊕M1(v). We shall show now that u is σ-finite. Arguing by contradiction, assume there is an uncountable family (uj)j∈Γ of mutually orthogonal non-zero tripotents in M with uj≤u for every j (see [17, Section 3]). Since uj∈M2(u) for every j and u⊥(w−v), it follows that uj⊥(w−v) for every j∈Γ. Arguing as above we obtain uj∈M2(v)⊕M1(v), for every j∈Γ. Having in mind that v is σ-finite, we can find a norm one functional ϕv∈M* whose support tripotent is v (see [17, Theorem 3.2]). By Lemma 3.3, ϕv gives rise to a norm ∥·∥ϕv on M2(v)⊕M1(v) defined by ∥x∥ϕv=(ϕv{x,x,v})1/2 ( x∈M2(v)⊕M1(v)). As uj is a non-zero element in M2(v)⊕M1(v) by the preceding paragraph, we obtain ϕv{uj,uj,v}=∥uj∥2>0. Therefore, there exists a positive constant Θ and an uncountable subset Γ′⊆Γ such that ϕv{uj,uj,v}>Θ for all j∈Γ′. Thus, for each natural m we can find j1≠j2≠⋯≠jm∈Γ′. Since the elements uj1,…,ujm are mutually orthogonal, we get 1=∥∑k=1mujk∥2≥∥∑k=1mujk∥ϕv2=ϕv{∑k=1mujk,∑k=1mujk,v}=∑k=1mϕv{ujk,ujk,v}>mΘ, which is impossible.□ To prove that Mσ is an inner ideal, we need another representation of M. To this end fix a complete tripotent e∈M. Applying [17, Theorem 3.4(ii)] we can find a family (eλ)λ∈Λ of mutually orthogonal σ-finite tripotents in M satisfying e=∑λ∈Λeλ. For each x∈M let us define Λx≔{λ∈Λ:L(eλ,eλ)(x)≠0}. Proposition 5.3. In the conditions above, Mσ={x∈M:Λxiscountable}and Mσis a norm-closed inner ideal of M. Proof Denote the set on the right-hand side by Mσ′. By the linearity of the Jordan product in the third variable, it follows that Mσ′ is a linear subspace. To show that it is an inner ideal, take x,z∈Mσ′ and y∈M. For each λ∈Λ⧹(Λx∪Λz), we deduce via Jordan identity, that L(eλ,eλ){x,y,z}={L(eλ,eλ)x,y,z}−{x,L(eλ,eλ)y,z}+{x,y,L(eλ,eλ)z}=−{x,L(eλ,eλ)y,z}. Moreover, since L(eλ,eλ)x=L(eλ,eλ)z=0, we get x,z∈M0(eλ). Since P0(eλ)y is in the 0-eigenspace of L(eλ,eλ) we have that L(eλ,eλ)(y)∈M1(eλ)⊕M2(eλ) and hence {x,L(eλ,eλ)(y),z}=0 by Peirce arithmetic. We have shown that Λ{x,y,z}⊆Λx∪Λz, and thus Λ{x,y,z} is countable, which proves that {x,y,z}∈Mσ′ and hence Mσ′ is an inner ideal of M. We continue by showing that Mσ⊂Mσ′. We shall first prove that Mσ′ contains all σ-finite tripotents in M. Let u be a σ-finite tripotent in M. We want to show that the set Λu is countable. We assume, on the contrary, that Λu is uncountable. Let ϕu∈M* be a norm one functional whose support tripotent is u. For every λ∈Λu, we have that eλ∈M0(u) because otherwise we would have L(eλ,eλ)(u)=0. Consequently, as in the proof of Proposition 5.2, we deduce that ϕu{eλ,eλ,u}>0. We can thus find a positive constant Θ and an uncountable subset Λu′⊆Λu such that ϕu{eλ,eλ,u}>Θ for all λ∈Λu′. As before, for each natural m we can find λ1≠λ2≠⋯≠λm∈Λu′. Then, applying the orthogonality of the elements eλj we get 1=∥∑j=1meλj∥2≥∥∑j=1meλj∥ϕu2=ϕu{∑j=1meλj,∑j=1meλj,u}=∑j=1mϕu{eλj,eλj,u}>mΘ, which gives a contradiction. This proves that Λu is countable, and hence u∈Mσ′. Let us now assume that x is any element of Mσ. Then its range tripotent, r(x), is σ-finite and hence r(x)∈Mσ′ by the previous paragraph. Since x∈M2(r(x)) is a positive and hence self-adjoint element, we have {r(x),x,r(x)}=x and hence x∈Mσ′ as Mσ′ is an inner ideal. This shows that Mσ⊂Mσ′. Conversely, let x∈Mσ′. In this case the set Λx is countable. The tripotent u=w*-∑λ∈Λxeλ is σ-finite in M, e=u+v, where v=w*-∑λ∈Λ⧹Λxeλ is another tripotent in M with u⊥v. Since {eλ,eλ,x}=0 for all λ∈Λ⧹Λx, it follows from the separate weak*-continuity of the triple product of M that {v,v,x}=0, that is, x∈M0(v). Hence also r(x)∈M0(v) (as M0(v) is a JBW*-subtriple of M). It follows that r(x)⊥v and hence r(x) is σ-finite by Proposition 5.2. We finally observe that, by Theorem 4.1, Mσ is a Σ-subspace and hence it is norm-closed (cf. Lemma 2.5(i)). This completes the proof.□ We are now ready to prove the main theorem of this section. Proof of Theorem 5.1 Mσ is a norm-closed inner ideal by Proposition 5.3. Let us prove the uniqueness. Let I be an inner ideal which is a 1-norming Σ-subspace. We will show that I contains all sigma-finite tripotents. Let e∈M be a sigma-finite tripotent, ϕ∈M* a normal functional of norm 1 such that e is the support tripotent of ϕ. By Lemma 2.5(v) there is x∈I of norm 1 with ϕ(x)=1. Further, we get r(x)∈I. Indeed, r(x) is contained in the weak*-closure of the JB*-subtriple of M generated by x. Since this subtriple is norm-separable, we get r(x)∈I by Lemma 2.5(i). In order to show e∈I, it is enough to show that e≤r(x). By (2.4), it is enough to prove that ϕ(r(x))=1. Proposition 2.5 in [45] assures that ϕ(x[12n+1])=ϕ(x)[12n+1]=1, for all natural n. Since ϕ is a normal functional and (x[12n+1])→r(x) in the weak* topology of M, it follows that ϕ(r(x))=1, as we desired. Now, if z∈Mσ is arbitrary, then there is a σ-finite tripotent f∈M with z∈M2(f). By the above we have f∈I. Since I is an inner ideal, we conclude that M2(f)⊂I, and hence z∈I. Therefore, Mσ⊂I. Lemma 2.5(iii) now shows that Mσ=I.□ Remark 5.4. It is possible to give a shorter proof of the fact that the predual of a JBW*-triple is 1-Plichko by using the main result of [5] at the cost of applying elementary submodels theory. However, this alternative argument does not yield Mσ as a concrete description of a Σ-subspace. We shall only sketch this variant: First, it is not too difficult to modify the decomposition (2.8) by writing M=(⨁j∈IAj⊗¯Gj)ℓ∞⊕ℓ∞N⊕ℓ∞pV, (5.1) where each Aj is a commutative von Neumann algebra, each Gj is a finite dimensional Cartan factor, p is a projection in a von Neumann algebra V, and N is a JBW*-algebra. Second, an almost word-by-word adaptation of the proof of [4, Theorem 1.1] shows that the predual of pV is 1-Plichko (compare Proposition 4.5). So is the predual of N by the main result of [5]. Finally, the summands Aj⊗¯Gj are seen to have 1-Plichko predual as in the proof of 4.6 (or by an easier argument using the finite dimensionality of Cj), and the stability of 1-Plichko spaces by ℓ1-sums ([34, Theorem 4.31(iii)] or Lemma 4.9) allows us to conclude. 6. The case of real JBW*-triples Introduced by J. M. Isidro et al. (see [32]), real JB*-triples are, by definition, the closed real subtriples of JB*-triples. Every complex JB*-triple is a real JB*-triple when we consider the underlying real Banach structure. Real and complex C*-algebras belong to the class of real JB*-triples. An equivalent reformulation asserts that real JB*-triples are in one-to-one correspondence with the real forms of JB*-triples. More precisely, for each real JB*-triple E, there exist a (complex) JB*-triple Ec and a period-2 conjugate-linear isometry (and hence a conjugate-linear triple isomorphism) τ:Ec→Ec such that E={b∈Ec:τ(b)=b}. The JB*-triple Ec identifies with the complexification of E (see [32, Proposition 2.2] or [9, Proposition 4.2.54]). In particular, every JB-algebra (and hence the self-adjoint part, Asa of every C*-algebra A) is a real JB*-triple. Henceforth, for each complex Banach space X, the symbol XR will denote the underlying real Banach space. In the conditions above, we can consider another period-2 conjugate-linear isometry τ♯:Ec*→Ec* defined by τ♯(φ)(z)≔φ(τ(z))¯(φ∈Ec*). It is further known that the operator (Ec*)τ♯→(Ecτ)*,φ↦φ∣E is an isometric real-linear bijection, where (Ec*)τ♯≔{φ∈Ec*:τ♯(φ)=φ}. A real JBW*-triple is a real JB*-triple which is also a dual Banach space ([32, Definition 4.1] and [43, Theorem 2.11]). It is known that every real JBW*-triple admits a unique (isometric) predual and its triple product is separately weak*-continuous (see [43, Proposition 2.3 and Theorem 2.11]). Actually, by the just quoted results, given a real JBW*-triple N there exists a JBW*-triple M and a weak*-to- weak* continuous period-2 conjugate-linear isometry τ:M→M such that N=Mτ. The mapping τ♯ maps M* into itself, and hence we can identify (M*)τ♯ with N*=(Mτ)*. We can also consider a weak*-continuous real-linear bicontractive projection P=12(Id+τ) of M onto N=Mτ, and a bicontractive real-linear projection of M* onto N* defined by Q=12(Id+τ♯). From now on, N, M, τ, P and Q will have the meaning explained in this paragraph. Due to the general lack for real JBW*-triples of the kind of structure results established by Horn and Neher for JBW*-triples in [30, 31], the proofs given in Section 4 cannot be applied for real JBW*-triples. Despite of the limitations appearing in the real setting, we shall see how the tools in previous section can be applied to prove that preduals of real JBW*-triples are 1-Plichko spaces too. We shall need to extend the concept of σ-finite tripotents to the setting of real JBW*-triples. The notions of tripotents, Peirce projections, Peirce decomposition are perfectly transferred to the real setting. The relations of orthogonality and order also make sense in the set of tripotents in N (cf. [32, 43]). Furthermore, for each tripotent e in N, Q(e) induces a decomposition of N into R-linear subspaces satisfying N=N1(e)⊕N0(e)⊕N−1(e), where Nk(e)≔{x∈N:Q(e)x=kx}, N2(e)=N1(e)⊕N−1(e)N0(e)=N1(e)⊕N0(e), {Nj(e),Nk(e),Nℓ(e)}⊂Njkℓ(e)ifjkℓ≠0,j,k,ℓ∈{0,±1},andzerootherwise. The natural projection of N onto Nk(e) is denoted by Pk(e). It is also known that P1(e), P−1(e), and P0(e) are all weak*-continuous. The subspace N1(e) is a weak*-closed Jordan subalgebra of the JBW-algebra (M2(e))sa, and hence N1(e) is a JBW-algebra. Given a normal functional ϕ∈N*, there exists a normal functional φ∈M* satisfying τ♯(φ)=φ and φ∣N=ϕ. Let e(φ) be the support tripotent of φ in M. Since 1=φ(e(φ))=φ(τ(e(φ)))¯=φ(τ(e(φ))), we deduce that τ(e(φ))≥e(φ). Applying that τ is a triple homomorphism, we get e(φ)=τ2(e(φ))≥τ(e(φ))≥e(φ), which proves that e(φ)=τ(e(φ))∈N. That is, the support tripotent of a τ♯-symmetric normal functional φ in M* is τ-symmetric. The tripotent e(φ) is called the support tripotent of ϕ in N, and it is denoted by e(ϕ). It is known that ϕ=ϕP1(e(ϕ)) and ϕ∣N1(e(ϕ)) is a faithful positive normal functional on the JBW-algebra N1(e(ϕ)) (compare [47, Lemma 2.7]). As in the complex setting, a tripotent e in N is called σ-finite if e does not majorize an uncountable orthogonal subset of tripotents in N. The real JBW*-triple N is called σ-finite if every tripotent in N is σ-finite. Proposition 6.1. In the setting fixed for this section, let ebe a tripotent in N. The following are equivalent: eis σ-finite in N; eis σ-finite in M; eis the support tripotent of a normal functional ϕin N*; eis the support tripotent of a τ♯-symmetric normal functional φ in M*.Consequently, for Nσ≔{x∈N:thereexistsaσ-finitetripotenteinNwith{e,e,x}=x}we have Nσ={x∈Mσ:τ(x)=x}=N∩Mσ,and the following are equivalent: Mis σ-finite (that is, Mσ=M); Nis σ-finite (that is, Nσ=N); Ncontains a complete σ-finite tripotent. Proof The implication (b)⇒(a) and the equivalence (c)⇔(d) are clear. The implication (d)⇒(b) follows from [17, Theorem 3.2]. To see (a)⇒(d), let us assume that e is σ-finite in N. Clearly e is the unit in the JBW-algebra N1(e), and since every family of mutually orthogonal projections in this algebra is a family of mutually orthogonal tripotents in N majorized by e, we deduce that e is a σ-finite projection in N1(e). [13, Theorem 4.6] assures the existence of a faithful normal state ϕ in (N1(e))*. By a slight abuse of notation, the symbol ϕ will also denote the functional ϕP1(e). Clearly ϕ∈N* and ϕ∣N1(e) is a faithful normal state. By the arguments above, there exists a τ♯-symmetric normal functional φ in M* such that φ∣N=ϕ. Let e(φ) be the support tripotent of φ in M. We have also commented before this proposition that τ(e(φ))=e(φ) (that is, e(φ)∈N) because ϕ is τ♯-symmetric. Since φ(e)=ϕ(e)=1, we deduce that e≥e(φ). Therefore, e(φ) is a projection in the JBW-algebra N1(e). Furthermore, ϕ(e(φ))=φ(e(φ))=1 and the faithfulness of ϕ∣N1(e) show that e=e(φ). This proves the equivalence of (a), (b), (c) and (d). The equality Nσ=N∩Mσ is clear from the first statement. Since a complete tripotent in N is a complete tripotent in M, the rest of the statement follows from the previous equivalences and [17, Theorem 4.4].□ We can prove now our main result for preduals of real JBW*-triples. Theorem 6.2. The predual of any real JBW*-triple Nis a 1-Plichko space. Moreover, N*is WLD if and only if Nis σ-finite. In the latter case N*is even WCG. Proof We keep the notation fixed for this section with N, M and τ as above. There exists a canonical isometric identification of MR with ((M*)R)*, where any x∈MR acts on (M*)R by the assignment ω↦Reω(x) ( ω∈(M*)R). Thus (M*)R is a real 1-Plichko space and Mσ is again a 1-norming σ-subspace by Theorem 4.1 and [36, Proposition 3.4]. In view of Lemma 2.6 to prove that the predual of N is 1-Plichko, it is enough to show that BN∩Mσ is weak*-dense in BN. Since Mσ is a 1-norming subspace we can easily see that BMσ is weak*-dense in BM. Take an element a∈BN⊂BM. Then there exists a net (aλ)⊂BMσ converging to a in the weak*-topology of M. Since τ is weak*-continuous and Mσ is a norm-closed τ-invariant subspace of M, we can easily see that (aλ+τ(aλ)2)→a in the weak*-topology of M, where (aλ+τ(aλ)2)⊂BNσ=BN∩Mσ, which proves the desired weak*-density. For the last statement, we observe that N is σ-finite if and only if M is (see Proposition 6.1), and hence the desired equivalence follows from Theorem 4.1 and the results presented in Sections 4 and 6. We also note that N σ-finite implies M σ-finite implies M* WCG implies N* WCG, being a complemented subspace.□ We can rediscover the following two results in [4] and [5] as corollaries of our last theorem. Corollary 6.3. ([4, Theorem 1.4]). Let Wbe a von Neumann algebra. Then the predual, (Wsa)*, of the self-adjoint part, Wsa, of Wis a 1-Plichko space. Moreover, (Wsa)*is WLD if and only if Wis σ-finite. In the latter case W*and (Wsa)*are even WCG. Corollary 6.4. ([5, Theorem 1.1]). 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