Diagonal automorphisms of the 2-adic ring C*-algebra

Diagonal automorphisms of the 2-adic ring C*-algebra Abstract The 2-adic ring C*-algebra Q2 naturally contains a copy of the Cuntz algebra O2 and, a fortiori, also of its diagonal subalgebra D2 with Cantor spectrum. This paper is aimed at studying the group AutD2(Q2) of the automorphisms of Q2 fixing D2 pointwise. It turns out that any such automorphism leaves O2 globally invariant. Furthermore, the subgroup AutD2(Q2) is shown to be maximal abelian in Aut(Q2). Saying exactly what the group is amounts to understanding when an automorphism of O2 that fixes D2 pointwise extends to Q2. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism. 1. Introduction As soon as the Cuntz algebras were introduced in [11], it was quickly realized that studying their endomorphisms and automorphisms would initiate a fruitful research season. The forecast could not possibly be more accurate, for after nearly 40 years they continue to be a major topic and a source of inspiration, as demonstrated by the recent literature that has been accumulating at an impressive rate, see for example [2–9]. Motivated by these works, we found that it is natural to ask ourselves whether such a case study would also provide the right tools to analyze the endomorphisms and automorphisms of other classes of C*-algebras, notably those recently associated with rings, fields and other algebraic objects. In fact, this was one of the main reasons why in [1] we started an investigation of the group Aut(Q2) of unital *-preserving automorphisms of the so-called dyadic ring C*-algebra of the integers Q2, a known C*-algebra (see for example [15] and the references therein) associated to the semidirect product semigroup Z⋊{1,2,22,23,…} that contains a copy of the Cuntz algebra O2 in a canonical way. Inter-alia, it was proved in [1] the useful fact that the canonical diagonal D2 maintains the property of being a maximal abelian subalgebra (MASA) in Q2 also. Actually, more is known, for D2 is even a Cartan subalgebra of both O2 and Q2 [13, Proposition 4.3]. The study there initiated is further developed in the present paper. In particular, our main focus is here on the structure of the set of those automorphisms of Q2 leaving the diagonal MASA D2 pointwise fixed, which will always be denoted by AutD2(Q2). Rather interestingly, this group turns out to be a maximal abelian subgroup of Aut(Q2). Moreover, we show that any of its elements restricts to an automorphism of O2 and it is indeed the unique extension of its restriction. It immediately follows from the analysis carried out in [10] that such restrictions are automorphisms of O2 induced by unitaries in D2, henceforth referred to as diagonal automorphisms for short. The results here obtained lend further support to the idea, already expressed in [1], that the group of automorphisms of Q2 is, in a sense, considerably smaller than that of O2, thus making it reasonable to ask the challenging question whether this group may be computed explicitly up to inner automorphisms. Indeed, we show that any extendible localized diagonal automorphism of O2 is necessarily the product of a gauge automorphism and a localized inner diagonal automorphism. The general case is still out of the reach of the techniques used in this paper instead. Even so, we do spot a necessary and sufficient condition for a diagonal automorphism to extend. Despite all our efforts to exploit the condition, to date this has not aided us in deciding whether the only extendible diagonal automorphisms of O2 are products of gauge and inner diagonal automorphisms. It is quite possible that the answer to this problem will also require to delve further into the fine ergodic properties of the odometer map. 2. Preliminaries and notations We recall the basic definitions and properties of the 2-adic ring C*-algebra so as to make the reading of the present paper suitable for a broader, not necessarily specialized, audience. This is the universal C*-algebra Q2 generated by an isometry S2 and a unitary U such that S2S2*+US2S2*U*=1 and S2U=U2S2. A very informative account of its most relevant properties is given in [15] as well as in our former work [1]. As far as the purposes of this work are concerned, it is important to mention that the Cuntz algebra O2, that is the universal C*-algebra generated by two isometries X1 and X2 such that X1X1*+X2X2*=1 can be thought of as a subalgebra of Q2 via the injective unital ∗-homomorphism that sends X1 to US2 and X2 to S2. As of now this ∗-homomorphism will always be understood without explicit mention, therefore, we simply write O2⊂Q2 to refer to the copy of O2 embedded in Q2 in this way. The 2-adic ring C*-algebra is actually a kind of a more symmetric version of O2, in which the Cuntz isometries S1 and S2 are now intertwined by the unitary U, to wit S2U=US1. This circumstance introduces a higher degree of rigidity, which is ultimately responsible for a shortage of outer automorphisms. Although not completely computed, the group Out(Q2) is nevertheless closely looked over in [1], where it is shown to be considerably smaller than Out(O2), while being still uncountable and non-commutative. To get a better idea of to what extent the former group is smaller than the latter, it is worthwhile to point up that endomorphisms or automorphisms of O2 will not in general extend to Q2, as widely discussed in [1]. Among those that do extend, there are the canonical endomorphism, the gauge automorphisms and the flip-flop. Of these only the first two will play an important role in this work. In particular, the gauge automorphisms will actually play an overriding role, which means they need a bit more exhaustive introduction. On O2, these are the automorphisms αθ acting on each isometry simply multiplying it by eiθ, that is αθ(Si)=eiθSi for i=1,2, where θ is any real number. The action of the one-dimensional torus T on O2 provided by the gauge automorphisms enables us to speak of the gauge invariant subalgebra F2⊂O2, which is by definition the C*-subalgebra whose elements are fixed by all the αθ’s. It is well known that F2 is the UHF algebra of type 2∞. Now the gauge automorphisms are immediately seen to extend to automorphisms αθ˜ of the whole Q2 with αθ˜(U)=U. Less obviously, each αθ˜ is an outer automorphism when it is not trivial, which is proved in [1]. Furthermore, the extended gauge automorphisms allow us to consider the gauge invariant subalgebra of Q2, which we denote by Q2T. Among other things, Q2T is known to be a Bunce–Deddens algebra. It is not particularly hard to prove that Q2T can also be described as the C*-subalgebra of Q2 generated from either F2 or D2 and U, where D2⊂F2 is a notable commutative subalgebra of the Cuntz algebra O2. Commonly referred to as the diagonal subalgebra, D2 is in fact the subalgebra generated by the diagonal projections Pα≐SαSα*, where for any multi-index α=(α1,α2,…,αk)∈⋃n{1,2}n the isometry Sα is the product Sα1Sα2…Sαk. The multi-index notation is rather convenient when making computations in Cuntz algebras and will be used extensively throughout the paper. In particular, we need to recall that ∣α∣ is the length of the multi-index α. By definition, the diagonal D2 is also the inductive limit of the increasing sequence of the finite-dimensional subalgebras D2k⊂D2k+1⊂D2 given by D2k≐span{Pα:∣α∣=k}, k∈N. That D2 is quite a remarkable subalgebra is then seen at the level of its spectrum, for the latter is the Cantor set K, of which many a concrete topological realization is known. However, in what follows we shall always think of it as the Tychonov infinite product {1,2}N. Not of less importance is the canonical endomorphism φ∈End(O2), which is defined on each element x∈O2 as φ(x)=S1xS1*+S2xS2*. By its very definition, it clearly extends to Q2, on which it still acts as a strongly ergodic map, namely ⋂nφn(Q2)=C1, as shown in [1]. The intertwining rules Six=φ(x)Si for any x∈Q2 with i=1,2 still hold true. In addition, the canonical endomorphism preserves the diagonal D2, acting on its spectrum as the usual shift map on {1,2}N. To complete the description of our framework, we still need to single out a distinguished representation of Q2 among the many, namely the so-called canonical representation [15]. This is the representation in which S2 and U are concrete operators acting on the Hilbert space ℓ2(Z) as S2ek=e2k and Uek=ek+1, for every k∈Z, where {ek:k∈Z} is the canonical basis of ℓ2(Z). Finally, in this representation, D2 can be seen as a norm-closed subalgebra of ℓ∞(Z), the diagonal operators with respect to the canonical basis. For any d∈ℓ∞(Z), we denote by d(k) its kth diagonal entries, that is d(k)≐(ek,dek), k∈Z. It is obvious that Ad(U) leaves ℓ∞(Z) invariant. It is slightly less obvious that it also leaves D2 globally invariant. Moreover, the spectrum of D2 is acted on by Ad(U) through the homeomorphism given by the so-called odometer on the Cantor set, which is known to be a uniquely ergodic map, see [12]. Finally, the action of Ad(U) on D2 is compatible with its inductive-limit structure, that is Ad(U)(D2k)=D2k for every k∈N. 3. General structure results on Aut2(2) Given an inclusion of C*-algebras A⊆B, we shall denote by Aut(B,A) the group of those automorphisms of B leaving A globally invariant, and by AutA(B) the group of those fixing A pointwise. Both endomorphisms and automorphism are tacitly assumed to be unital whenever our C*-algebras are unital, as D2 and Q2 are. Having a general content, the following result may possibly be known, cf. [10] for instance. Because we have no explicit reference, a precise statement is nevertheless included along with its proof. To state it as clearly as possible, however, we still need to set some notations. In particular, we recall that if H is any subgroup of a group G, its normalizer is the largest subgroup NH(G) in which H is contained as a normal subgroup. More explicitly, NH(G) can be identified with the set of those g∈G such that gHg−1=H. Finally, if S is any subset of Aut(B), we denote by BS the sub- C*-algebra of B whose elements are fixed by all automorphisms of S. Here follows the result. Proposition 3.1. Let A⊆Bbe a unital inclusion of C*-algebras. Then Aut(B,A)⊆NAutA(B)(Aut(B)); If BAutA(B)=Aone has Aut(B,A)=NAutA(B)(Aut(B)); If Ais a MASA in B, then BAutA(B)=Aand Aut(B,A)=NAutA(B)(Aut(B)). Proof For the first property, let γ∈Aut(B,A) and β∈AutA(B). Since γ◦β◦γ−1(a)=a for all a∈A, we get that γ belongs to NAutA(B)(Aut(B)). For the second, consider γ∈NAutA(B)(Aut(B)) and β∈AutA(B). We have that γ◦β◦γ−1(a)=a for all a∈A, hence β◦γ−1(a)=γ−1(a). This means that γ−1(a)∈BAutA(B) which is by hypothesis equal to A, thus γ∈Aut(B,A). Finally, for the third, we observe that any u∈U(A) gives rise to an element of AutA(B), namely Ad(u). This means that for any x∈BAutA(B) we have uxu*=x. Since A is a MASA, then x∈A and the second property in turn implies that Aut(B,A)=NAutA(B)(Aut(B)).□ By applying the former result to the inclusion D2⊂Q2, the set equality Q2AutD2(Q2)=D2 is immediately got to. Now AutD2(Q2) contains both {αθ˜∣θ∈R} and {Ad(u),u∈U(D2)}, where αθ˜ is the unique extension to Q2 of the gauge automorphism αθ∈Aut(O2), see [1]. Furthermore, the intersection AutD2(Q2)⋂Inn(Q2) is easily seen to reduce to {Ad(u),u∈U(D2)} thanks to maximality of D2 again. Since D2 is globally invariant under Ad(U), for any α∈AutD2(Q2) we have α(UdU*)=UdU*=α(U)dα(U)*foralld∈D2, that is U*α(U) commutes with every d∈U(D2) and therefore by maximality U*α(U)=đα for some đα∈U(D2), which we rewrite as α(U)=Uđα. It is also clear that α−1(U)=Uđα*. To take but one example, when α is an inner automorphism, say Ad(u) for some u in U(D2), the corresponding đAd(u) is nothing but U*uUu*. More importantly, the map α↦đα is easily recognized to be a group homomorphism between AutD2(Q2) and U(D2). Moreover, its kernel coincides with AutQ2T(Q2), cf. Proposition 3.4. Our next goal is to show that the Cuntz–Takesaki unitary uα≐α(S1)S1*+α(S2)S2* belongs to D2 as well. Proposition 3.2. Let αbe in AutD2(Q2). Then the corresponding unitary uα lies in D2. Proof First we observe that α(Si)=uαSi for i=1,2. By maximality of D2⊂Q2, it is enough to prove that uα commutes with the generating projections Pi1i2…ik of D2. This can be easily seen by induction on k, as done by Cuntz for O2. The case of length one reduces to the computation Pi=α(Pi)=α(SiSi*)=uαSiSi*uα*=uαPiuα*. The case of length two entails the computation Pij=α(Pij)=α(SiSjSj*Si*)=uαSi(uαSjSj*uα*)Si*uα*=uαSiSjSj*Si*uα*=uαPijuα*. It is now clear how to go on.□ The following result can be derived at once from the foregoing proposition. In this respect, it is worth recalling the fact, proved in [10], that there exists an explicit group isomorphism between U(D2) and AutD2(O2) given by d↦λd, where λd(Si)=dSi, i=1,2, already showing the abelianness of AutD2(O2). Corollary 3.3. Any α∈AutD2(Q2)restricts to an automorphism of the Cuntz algebra O2fixing pointwise the diagonal D2and it is the unique extension of such restriction. In particular, the group AutD2(Q2)is abelian. Proof It is clear from the previous proposition that α(Si)∈O2 for i=1,2, so that α(O2)⊆O2. Now, uα*Si∈O2 and α(uα*Si)=uα*α(Si)=uα*uαSi=Si, i=1,2, thus showing that indeed α(O2)=O2. We conclude that α is an extension to Q2 of its restriction to O2, and the statement about uniqueness follows at once from the rigidity result, proved in [1, Section 4], that two automorphisms of Q2, coinciding on O2, must be the same. For the last claim, if αi∈AutD2(Q2), i=1,2, we compute α1(α2(U))=α1(Uđα2)=Uđα1đα2=Uđα2đα1=α2(α1(U)) and, similarly, α1(α2(S2))=α1(uα2(S2))=uα2uα1S2=α2(α1(S2)). The conclusion readily follows.□ Every element in AutD2(Q2) can thus be written as the unique extension of an element λd∈AutD2(O2), for some d∈U(D2). We denote such extension as λd˜. However, one should not expect all automorphisms λd with d∈U(D2) to extend to Q2. Denoting by U˜(D2) the set of all d∈U(D2) such that λd is extendible to an automorphism of Q2, it is then easy to deduce from the above discussion that U˜(D2) is actually a group (a subgroup of U(D2)), and there exists a group isomorphism between U˜(D2) and AutD2(Q2) given by d↦λd˜. Thus far we have seen that any α∈AutD2(Q2) acts on U as α(U)=Uđα for some đα∈D2. Now it is also possible to rewrite this relation in the form α(U)=đˇαU, where đˇα is simply given by UđαU* and is still a unitary of D2. As α=λd˜, we can simply write dˇ instead of đˇα=đˇλd˜ for d∈U˜(D2). For the same reason as above, the map d↦dˇ is a group homomorphism from U˜(D2) to U(D2). In fact, this map will turn out to be vital in the next sections. Contrary to what one might expect, though, it has been proved to be a difficult task to establish a priori whether it is norm continuous, possibly because determining its domain U˜(D2) is just another way to recast our main problem. Nevertheless, its kernel can be described quite explicitly. Proposition 3.4. The kernel of the map d↦dˇ (defined on U˜(D2)) is the subgroup of the gauge automorphisms. Actually, one has AutQ2T(Q2)=AutF2(Q2)={αθ˜:θ∈R}. Proof Clearly the condition dˇ=1 is the same as λd˜(U)=U. Since Q2T coincides with C*(U,D2), see for example [1, Section 2], it means that λd˜∈AutQ2T(Q2) so that λd∈AutO2T(O2). The conclusion now readily follows from the fact that the automorphisms of O2 fixing the canonical UHF subalgebra F2=O2T pointwise are precisely the gauge automorphisms, as proved by Cuntz in [10].□ In particular, the restriction map AutD2(Q2)∋λ→λ↾O2∈AutD2(O2) induces a group embedding which allows us to think of the former group as a subgroup of the latter. Therefore, as of now we will simply write AutD2(Q2)⊂AutD2(O2) to mean that. Of course the inclusion is proper. In other words, not all the automorphisms of O2 that leave the diagonal D2 globally invariant will extend. As a matter of fact, very few automorphisms can be extended. Although we do not have a general explicit description of all extendible automorphisms yet, we do have a complete description for a particular class of automorphisms. This is just the subgroup AutD2(O2)loc of those localized automorphisms we mentioned above in passing. Actually, the terminology comes from Quantum Field Theory. Roughly speaking, an automorphism is localized when it preserves the union of the matrix subalgebras. More precisely, an automorphism λu∈Aut(On) is said to be localized when the corresponding unitary u∈U(On) belongs to the algebraic dense subalgebra ⋃kFkn⊂On, where Fkn is generated by the elements of the form SαSβ* with α,β∈{1,…,n}k. Furthermore, the inclusion AutD2(Q2)⊂AutD2(O2) allows us to define a subgroup AutD2(Q2)loc as the intersection AutD2(Q2)⋂Aut(O2)loc. As maintained in the abstract, we will prove that AutD2(Q2)loc is so small that the sole localized automorphisms fixing D2 that extend are the composition of a localized inner automorphism with a gauge automorphism. Before going on with our discussion, we would like to point out a remark for the sake of completeness. Remark 3.5. Let d∈U(D2) and consider the associated automorphism λd of O2. If λd extends to an endomorphism λ of Q2, then λ is actually an automorphism, that is d∈U˜(D2) and λ=λd˜. Indeed, λ(Q2) contains λd(O2)=O2. Moreover, λ(U)=d˜U for a suitable d˜∈U(D2) (the same argument as for automorphisms), so that λ(d˜*U)=d˜*d˜U=U. All in all, the extension is nothing but λd˜ (and d˜=dˇ). Going back to AutD2(Q2), we have shown that it is abelian, but we want to improve our knowledge by proving that it is also maximal abelian in Aut(Q2), in a way that closely resembles what happens for the Cuntz algebra O2 [10]. Here follows the proof. Theorem 3.6. The subgroup AutD2(Q2)is maximal abelian in Aut(Q2). Proof Let α be an automorphism of Q2 that commutes with AutD2(Q2). In particular [α,Ad(u)]=0 for every u∈U(D2), to wit Ad(α(u))=Ad(u). As the center of Q2 is trivial, we see that α(u)=χ(u)u for every u∈U(D2), where χ is a character of the group U(D2). Our result will be proved once we show χ(u)=1 for every u∈U(D2). To this aim, note that the equality α(u)=χ(u)u says that D2 is at least globally invariant under the action of α. With a slight abuse of notation, we still denote by α the restriction of α to D2≅C(K), where K is the Cantor set. Let Φ∈Homeo(K) such that α(f)=f◦Φ for every f∈C(K). The identity obtained above is then recast in terms of Φ as f◦Φ=χ(f)f for every f∈C(K,T). We claim that χ(U(D2))⊂T is at most countable. If so, the theorem can now be easily inferred. Indeed, if Φ is not the identity map, then there exists x∈K such that Φ(x)≠x. Then pick a function f∈C(K,T) such that f(x)=1 and f(Φ(x))=eiθ. The equality f◦Φ=χ(f)f evaluated at x gives χ(f)=eiθ, that is χ is onto T. To really achieve the result we are thus left with the task of proving the claim. This should be quite a standard fact from ergodic theory. However, we do give a complete proof. If μ is any Borel Φ-invariant measure on K, we can consider the Hilbert space L2(K,μ), which is separable because K is metrizable, and the Koopman unitary operator Uϕ associated with Φ, whose action is simply given by UΦ(f)=f◦Φ a.e. for every f∈L2(K). As eigenfunctions of UΦ associated with different eigenvalues are orthogonal and χ(f) is an eigenvalue for every f∈C(K,T), we see that {χ(f):f∈C(K,T)} is a countable set by virtue of separability.□ Remark 3.7. We can also provide an alternative argument for the above result, proving more directly that α(Pβ)=Pβ for all the multi-indices β. First of all we observe that the relation α(d)=χ(d)d implies that the spectrum of the unitary d is invariant under the rotation of χ(d). We begin with the case of P1. Consider the unitary d1=P1+ei2πθP2 with θ≠±1. On the one hand, we know that α(d1)=χ(d1)d1. On the other hand, since the spectrum of d1 is not invariant under non-trivial rotations, we find that χ(d1) must be 1. The same reasoning applies to the unitary d˜1=P1−eiθP2 too, and so we get the equality χ(d˜1)=1, hence α(P1)=α(d1+d˜12)=P1. We now deal with the general case of a Pβ with β being a multi-index of length k in much the same way. Consider the two unitary operators dβ=∑∣γ∣=k,γ≠βeiθPγ+Pβ and d˜β=−∑∣γ∣=k,γ≠βeiθPγ+Pβ. By the same argument as above we still find both α(dβ)=dβ and α(d˜β)=d˜β, and thus α(Pβ)=α(dβ+d˜β2)=Pβ and we are done. 4. Necessary and sufficient conditions for extendability Thanks to the results achieved in the last section, giving a complete non-tautological description of AutD2(Q2) entails studying those unitaries d∈U(D2) for which the corresponding λd∈Aut(O2) may be extended to Q2. This section is mainly concerned with problems of this sort. When an automorphism λd extends, we will say every so often that the corresponding d is extendible itself. This is undoubtedly a slight abuse of terminology, but it aids brevity. Here follows our first result. Lemma 4.1. Let dbe in U(D2). Then λd∈Aut(O2)extends to an endomorphism of Q2if and only if there exists a d˜in U(D2)such that d˜UdS1=dS2d˜U (4.1) d˜UdS2=dS1. (4.2)Moreover, such an extension is automatically an automorphism whenever it exists, that is d∈U˜(D2), and d˜=dˇ. Proof If λd extends, then the two equalities in the statement are easily verified with d˜=dˇ if one applies its extension λd˜ (cf. Remark 3.5) to US1=S2U and US2=S1, respectively, also taking into account that λd˜(U)=dˇU. The converse is dealt with analogously by noting that the pair ( d˜U,dS2) in Q2 still satisfies the defining relations of Q2 and, therefore, by universality, there exists an endomorphism λ of Q2 such that λ(S2)=dS2 and λ(U)=d˜U. But then, λ(S1)=λ(US2)=d˜UdS2=dS1 by Equation (4.2), so that λ extends λd.□ At this point, the reader may be wondering whether it ever happens that d=dˇ. In fact, it turns out that this is never the case unless d=1, namely we have the following result. Proposition 4.2. The unitary d=1is the unique fixed point of the map U˜(D2)∋d↦dˇ∈U(D2). Proof If we work in the canonical representation, we simply need to show that d(k)=1 for every k∈Z. We first handle the even entries. Formula (4.2) becomes dUdS2=dS1, which in turn gives dS2=U*S1=S2. Now by computing the above equality on the vectors of the canonical basis of ℓ2(Z) we get d(2k)=1 for all k∈Z. As for the odd entries, Formula (4.1) leads to dUdS1=dS2dU, which yields d(2k+1)=d(k+1) for all k∈Z. This in turn says all odd entries of d are 1 as well, apart from d(1), which is in fact not determined by this condition. However, it cannot be different from 1, for otherwise d would not even belong to D2.□ Although more focused on the Cuntz algebra O2, the next useful result is included all the same. In fact, we do believe that it may shed some light on applications yet to come. Recall that if d,d′∈U(D2), then Ad(d′)◦λd=λd′dφ(d′)*. In particular, taking d′=d* we get that λd is extendible if and only if λφ(d) is extendible. Proposition 4.3. Let λd∈Aut(O2)be an extendible automorphism. Then either λdis a gauge automorphism or λdUd*U*is outer. Proof First of all we prove that (4.3) To this aim, rewrite Formula (4.2) as dˇUdS2=dˇUdU*S1=dS1. Then, by using the identity =dUd*U* and the multiplicativity of the map ˇ one finds (4.4) whence the claim. Likewise, Formula (4.1) yields dˇUdS1=dˇUdU*S2U=dS2dˇU=dφ(dˇ)S2U and thus dˇUdU*S2=dφ(dˇ)S2, that is dˇd*UdU*φ(dˇ*)S2=S2. (4.5) The former equality actually shows that the automorphism Ad(dˇ*)◦λdUd*U* fixes S2, that is Ad(dˇ*)◦λdUd*U*[S2]=S2. By [16, Corollary B], then either Ad(dˇ*)◦λdUd*U* is the identity or Ad(dˇ*)◦λdUd*U* is an outer automorphism of O2. In the first case dˇφ(dˇ)*=dUd*U*= = hence By applying the last equality to (4.3), we get which proves that dˇ=1, thus λd is a gauge automorphism by Proposition 3.4. In the second case, clearly λdUd*U* is an outer automorphism of O2.□ At any rate, a first application can be given at once. Corollary 4.4. If λd, d∈U(D2)is a non-trivial inner automorphism of O2, then λUdU*is outer. We can now resume to our general discussion. With this in mind, we start by spotting a useful necessary condition on d for the corresponding λd to extend. Proposition 4.5. Let dbe a unitary in D2. If λdextends to Q2, then d(0)=d(−1). Proof Owing to the extendability of λd the isometries dS1=λd(S1) and dS2=λd(S2) are still intertwined, that is they are unitarily equivalent. In particular, their point spectra must coincide. Now the equalities σp(dS1)={d(−1)} and σp(dS2)={d(0)} are both easily checked, hence the thesis follows.□ It goes without saying that the condition is only necessary. Even so, it does have the merit of highlighting a property of which we will have to make an extensive use. Therefore, unless otherwise stated, our unitaries d∈U(D2) will always satisfy the condition d(0)=d(−1). In addition, there is no lack of generality if we further assume that both d(0) and d(−1) equal 1. For if this were not the case, we could always multiply λd by a suitable gauge automorphism, which of course would not affect the extendability of λd, since gauge automorphisms certainly extend to Q2. Finally, we will work in the canonical representation and adopt the Dirac bra-ket notation for rank-one operators: for any given u,v∈ℓ2(Z) the operator w→(v,w)u is denoted by ∣u⟩⟨v∣. This said, we can now state a result which sheds further light on the relation between d and dˇ by relating our setup to some findings in [15]. Theorem 4.6. Let dbe a unitary in D2such that d(0)=d(−1)=1; then there exists the strong limit d∞of the sequence dk≐dφ(d)φ2(d)…φk−1(d)in the canonical representation; λdis weakly inner in the canonical representation restricted to O2; λdextends to a representation of Q2on ℓ2(Z); the automorphism λd∈Aut(O2)extends to an automorphism of Q2if and only if, for a unique α∈T, the strong limit of the sequence xk≐α∣e0⟩⟨e0∣+dφ(d)φ2(d)…φk−1(d)(∑i=0k−1S2iS1S1*(S2*)i)Uφk−1(d*)…φ(d*)d*U*belongs to D2, in which case the limit coincides with dˇand α=dˇ(0); if λdextends to an automorphism λd˜, then λd˜is weakly inner in the canonical representation if ( λdis inner or) dˇ(0)=1. Proof To begin with, we note that for every j∈Z the sequence {dk(j):k∈N}⊂T is eventually constant, as we certainly have dk(j)=d∣j∣(j) for all k≥∣j∣ thanks to φ(d)(k)=d([k/2]), k∈Z and d(0)=d(−1)=1. In particular, this shows that {dk:k∈N} is strongly convergent to a unitary d∞∈ℓ∞(Z). Actually, the operator d∞ thus exhibited implements λd, that is dSi=d∞Sid∞*, i=1,2. Indeed, d∞Sid∞*=limkdkSidk*=limkdkφ(dk)*Si=limkdφk(d*)Si=dSi, (4.6) where we have used the equalities Six=φ(x)Si, for all x∈O2 and i=1,2 and that φk(d) strongly converges to 1. Because Ad(d∞) restricts to O2 as λd, it also restricts to Q2 if only as a representation of the latter algebra on the Hilbert space ℓ2(Z). To ease some of the computations we need to make, it is now particularly convenient to introduce the sequence of projections Qk≐∣e0⟩⟨e0∣+∑i=0k−1S2iS1S1*(S2*)i, which act on ℓ2(Z). It is then not difficult to see that now the xk’s in the statement take on the much simpler form xk=(α−1)∣e0⟩⟨e0∣+QkdkUdk*U*. Now Qk strongly converges to the identity I, as shown by straightforward computations. Furthermore, the sequence Udk*U* converges too, since dk does. This shows that xk is strongly convergent to a limit δ≐(α−1)∣e0⟩⟨e0∣+d∞Ud∞*U*. Note that δ is a unitary lying in ℓ∞(Z)=D2″, for the equality see [1, Section 2]. We also point out the equality dk=dφ(dk−1), k∈{2,3,…}∪{∞}, which is necessary to carry out some of the following computations. We can now deal with 4. We start with the if part. In view of Lemma 4.1, it is enough to show that S2˜U˜≐U˜2S2˜ and S1˜≐U˜S2˜, where U˜=δU∈Q2 and Si˜=dSi. As for the first equality, we rewrite δU as δU=(α−1)∣e0⟩⟨e−1∣+d∞Ud∞*, where ∣e0⟩⟨e−1∣(v)=v(−1)e0 for all v∈ℓ2(Z). If we now use the expression obtained above, we can compute U˜2S2˜ as δUδUdS2=((α−1)∣e0⟩⟨e−1∣+d∞Ud∞*)2dS2=(α−1)∣e0⟩⟨e−1∣d∞Ud∞*dS2+d∞U2d∞*dS2=(α−1)∣e0⟩⟨e−1∣+d∞U2d∞*dS2, since ∣e0⟩⟨e−1∣d∞Ud∞*dS2ek is always zero unless k=−1, in which case it is equal to e0, while dS2δU=dS2((α−1)∣e0⟩⟨e−1∣+d∞Ud∞*)=(α−1)∣e0⟩⟨e−1∣+dS2d∞Ud∞*, since dS2e0=e0. It remains to show that d∞U2d∞*dS2=dS2d∞Ud∞*. However, these two terms are equal to d∞U2S2d∞* and d∞S2Ud∞*, respectively, where we used equation (4.6). The second equality is dealt with by means of a still easier computation: δUdS2=((α−1)∣e0⟩⟨e−1∣+d∞Ud∞*)dS2=d∞Ud∞*dS2=d∞S1d∞*=dS1. It is now clear that dˇ=(α−1)∣e0⟩⟨e0∣+d∞Ud∞*U*∈Q2 and it follows at once that dˇ(0)=α. For the only if part, if we think of λd as a representation of O2 on ℓ2(Z), the conclusion follows by the uniqueness pointed out in [15, Remark 4.2] (applied to dˇU). Finally, the condition dˇ(1)=1 leads to the equality dˇ=d∞Ud∞*U*, whence λd˜(U)=dˇU=d∞Ud∞*=Ad(d∞)(U), which says λd˜ is implemented by d∞.□ Needless to say, if λd is inner then (it extends to Q2 and) its extension in still inner. Conversely, if λd extends to an inner automorphism of Q2, then λd itself must be inner by the maximality of D2 in Q2 [1, Section 3.1]. Note that if λd=Ad(d′), with d′∈U(D2) then dˇ(0)=d′(0)d′(−1)¯. If moreover dˇ(0)=1, then d′ is nothing but d∞, which is thus in D2. Finally, it is not difficult to realize that the above theorem could also be set in other representations. Nevertheless, we shall refrain from discussing this issue any further, not least because it goes beyond the scope of the present work. For the results in the next section, we need to take a closer look at inner diagonal automorphisms, to which the rest of this section is addressed. To do that, we first prove a general lemma. Lemma 4.7. For any d∈U(D2)the sequence (S2*)ndS2nconverges in norm to the scalar d(0)1. Proof Given any ε>0 we can pick an algebraic dk∈D2k such that ∥d−dk∥<ε2. As ∥(S2*)ndS2n−d(0)1∥≤∥(S2*)n(d−dk)S2n∥+∥(S2*)ndkS2n−d(0)1∥≤ε2+∥(S2*)ndkS2n−d(0)1∥. So the conclusion is obtained if we can also prove that ∥(S2*)ndkS2n−d(0)1∥ may be made as small as wished. To this aim, it is enough to write dk as a finite linear combination of projections, say dk=∑∣α∣=kdαPα, where all the coefficients dα are in T. Thought of as a continuous function on the Cantor set, the unitary dk is nothing but dk(x)=∑∣α∣=kdαχα(x), and so there holds the inequality supx∈{1,2}N∣d(x)−∑∣α∣=kdαχα(x)∣<ε2. In particular, we also have ∣d({222…})−d22…2∣=∣d(0)−d22…2∣<ε2. But as soon as n is greater than k, the product (S2*)ndkS2n reduces to d22…21, and the conclusion is thus proved, being more exactly ∥(S2*)ndS2n−d(0)1∥≤ε.□ Remark 4.8. The same conclusion is also got to in the canonical representation. Indeed, since (S2*)ndS2nek=d(k2n)ek for any k∈Z, the claim amounts to proving that limnd(k2n)=d(0) for any d∈D2, with the limit being uniform in k. This is again easily proved by approximating in norm any such d with algebraic unitaries as closely as necessary, which is much the same as in the proof above. In our next result, there appear infinite products in a C*-algebraic framework. Since this is a topic seldom discussed in the literature, some comments as to which sense the convergence is understood in are necessary to dispel any possible doubt. The most refined notion one could work with essentially dates back to von Neumann, and is the one intended with respect to the direct net of finite subsets of N ordered by inclusion. However, we shall not need to be that demanding. For our purposes, we may as well make do with the usual notion that an infinite product ∏ai converges (in norm) if the sequence pn≐∏i=1nai does. Proposition 4.9. Let dbe a unitary in D2such that d(−1)=d(0)=1. Consider the following claims: The automorphism λdis inner. The automorphism λdextends to Q2and the infinite product ∏i=1∞(S2*)idS2iconverges in norm to a unitary in D2. There exists a unitary d′∈D2such that λd(S2)=d′S2d′*.Then we have the chain of implications 1⇒2⇔3. Moreover, when λdis inner the above infinite product converges to d′*, with d′being the unique unitary in D2such that d′(0)=1and d′φ(d′)*=das well as satisfying λd(S2)=d′S2d′*. Proof For the implication 1⇒2, we only need to prove that the infinite product converges. To this aim, let d′∈U(D2) such that d′(0)=1 and d′φ(d′)*=d. By means of a simple computation by induction, we gain the formula ∏i=1n(S2*)idS2i=(S2*)nd′S2nd′*. So the conclusion is arrived at by a straightforward application of the former lemma. For 2⇒3, we begin observing that it can be easily seen that ∏j=1n(S2*)jdS2j=(S2*)ndnS2n. By the following computations, we get (∏j=1n(S2*)jd*S2j)S2(∏j=1n(S2*)jdS2j)=((S2*)ndn*S2n)S2((S2*)ndnS2n)=(S2*)ndn*S2[S2n(S2*)n]dnS2n=(S2*)ndn*S2dnS2n=(S2*)ndn*φ(dn)S2S2n=(S2*)nd*φn(d)S2n+1=((S2*)nd*S2n)dS2. Therefore, it follows that limn(∏j=1n(S2*)jd*S2j)S2(∏j=1n(S2*)jdS2j)=limnd(S2*)nd*S2nS2=d(limn(S2*)nd*S2n)S2=dS2. So if we denote by d′ the limit of ∏j=1n(S2*)jd*S2j, we have the equality d′S2d′*=dS2. Finally the implication 3⇒2 is dealt with by similar computations as in the proof of 1⇒2.□ A well-known sufficient condition for the product in point 2 to exist (even with respect to the von Neumann notion of convergence) is that ∑i=1∞∥1−(S2*)idS2i∥<∞. 5. Localized extendible automorphisms While chiefly devoted to presenting our solution to the extension problem for localized automorphisms, this section also includes some results on general, that is possibly non-localized, unitaries. In fact, findings of this sort are interesting in their own right as well as being necessary tools to attack the case of localized automorphisms. First, we prove a structural result about the diagonal elements with the extension property. Theorem 5.1. If d∈U(D2)is an extendible unitary, then d, dˇand Usatisfy the following identity: d=UdU*dˇ(S1S1*+φ(dˇ)*S2S2*). (5.1) Proof The claim follows as in the proof of Proposition 4.3. Indeed, dS1=dˇUdU*S1dS2=dˇφ(dˇ*)UdU*S2 and we are done.□ Now a handful of corollaries can be easily derived. Corollary 5.2. Let λd, d∈U(D2), be an extendible automorphism. Then dˇ=dˇ(0)∏i=1∞(S2*)id*UdU*S2i. Proof By Formula (5.1), we have that ∏i=1k(S2*)idUd*U*S2i=∏i=1k(S2*)i(ď(S1S1*+φ(ď)*S2S2*))S2i=∏i=1k(S2*)i(ďφ(ď)*S2S2*)S2i=∏i=1k(S2*)i(ďφ(ď)*)S2i=(S2*)k(ďφ(ď)*)kS2k=(S2*)kďφk(ď)*S2k=(S2*)kďS2kď*, which converges in norm to dˇ*dˇ(0) as k→+∞ by Lemma 4.7.□ Corollary 5.3. Let λd, d∈U(D2), be an extendible automorphism. Then the product ∏i=1∞(S2*)idS2isits in D2if and only if ∏i=1∞(S2*)iUdU*S2idoes. Proof By formula (5.1) we have that ∏i=1k(S2*)idS2i=∏i=1k(S2*)i[dˇUdU*(S1S1*+φ(dˇ)*S2S2*)]S2i=∏i=1k(S2*)i[dˇφ(dˇ)*UdU*S2S2*]S2i=∏i=1k(S2*)i[dˇφ(dˇ)*UdU*]S2i=(∏i=1k(S2*)idˇφ(dˇ)*S2i)(∏i=1k(S2*)iUdU*S2i). Since the sequence ∏i=1k(S2*)idˇφ(dˇ)*S2i is norm-convergent as k→+∞ we have the claim.□ The former corollary applies in particular to localized unitaries. But in this case, only finitely many terms occur in the infinite product that gives the corresponding dˇ. More precisely, if d∈U(D2k) then dˇ=dˇ(0)∏i=1k−1(S2*)id*UdU*S2i. As a result, the following corollary is proved forthwith. Corollary 5.4. Let kbe a non-negative integer and let d∈U(D2k)be such that λdis extendible. Then dˇ∈U(D2k−1). This simple yet effective remark enables us to prove a preliminary result. Proposition 5.5. Let b∈U(D2k)be such that λbis extendible and bS2=S2. Then b=1. Proof We proceed by induction. The case k=1 follows immediately from Proposition 4.5. Now let us suppose that b∈U(D2k) with k>1. Notice that one must have b=bS1S1*+S2S2* and, moreover, bS1=λb(S1)=λb(US2)=bˇUS2=bˇS1, thus giving b=bˇS1S1*+S2S2*. By Corollary 5.4, the unitary bˇ, and hence b, is in U(D2k−1), and we are done.□ We are now in a position to prove the main result of the section. Theorem 5.6. Let d∈U(D2k)be such that λdis extendible. Then λdis the product of a gauge automorphism and an inner localized automorphism. Proof As explained, we can suppose without loss of generality that d(0)=d(−1)=1. Proposition 4.9 yields a d′ in U(D2k−1) such that Ad(d′*)◦λd(S2)=S2. The conclusion readily follows by Proposition 5.5 applied to b=d′*φ(d′)d∈U(D2k).□ At this point, it is worthwhile to stress that the proof of the main theorem also shows that for localized automorphisms the three conditions in Proposition 4.9 are actually equivalent. Indeed, whenever d is localized the infinite product in 2. does converge being a finite product, and thus the unitary in 3. is localized too. There is yet another consequence of Proposition 5.5 that deserves to be highlighted as a separate statement. Corollary 5.7. The groups AutD2(Q2)locand AutC*(S2)(Q2)have trivial intersection. One might wonder whether the subscript loc may be got rid of altogether in the last statement. Remark 5.8. The above discussion implies that if dn,d∈U(D2) are such that λdn and λd are extendible to Q2 and ∥dn−d∥→0 as n→∞, then one has the continuity property ∥dnˇ(S1S1*+φ(dnˇ)*S2S2*)−dˇ(S1S1*+φ(dˇ)*S2S2*)∥→0 as n→∞. If we multiply by S1S1*, we see that ∥dˇnS1−dˇS1∥ goes to zero as well. In order to prove the continuity of the map d→dˇ with respect to the norm topology, we should also check limn∥dˇnS2−dˇS2∥=0. As a matter of fact, all we can prove is the convergence only holds strongly. This is indeed a consequence of a general fact applied to {dˇ*dnˇ}: for any sequence {un}⊂U(D2) such that both limn∥unS1−S1∥ and limn∥unS2un*−S2∥ are zero, one also has limn∣un(2k)−1∣=0, for every k∈Z. However, the limit will in general fail to be uniform in k.This is because we can only say ∣un(2k)−1∣ is less than the sum ∣un(2k)−un(k)∣+∣un(k)−un(k2)∣+…+∣un(k2h)−1∣, with h being ≐min{l:k2lisodd}, and so the number of terms occurring in the sum depends upon what k is (at worst, when k=2l, exactly l terms are needed). 6. Localizing dˇ There is a last property of the group homomorphism ˇ that is well worth a thorough discussion. We have already observed that dˇ is localized whenever d is. Moreover, its kernel is obviously made up of localized unitaries. This seems to indicate that the other way around might also hold true, namely that an extendible d∈U(D2), whose dˇ is localized, has to be localized itself. This is quite the situation we are in. Far from immediate, the proof requires some preliminary work to do instead. More precisely, we need to give a closer look at the image of our homomorphism. To do so, we point out that between the entries of d and dˇ with respect to the canonical basis {ek}k∈Z in the canonical representation on ℓ2(Z) there hold some relations. More precisely, by evaluating (4.1) and (4.2) on ek−1 and ek, respectively, we get the following couple of equations: d(2k−1)dˇ(2k)=dˇ(k)d(2k) (6.1) dˇ(2k+1)d(2k)=d(2k+1) (6.2) to be satisfied by every integer k. Furthermore, the former equation can be recast in terms of the canonical endomorphism φ as dˇ*φ(dˇ)S2S2*=d*UdU*S2S2*. Here below follows a technical result. Lemma 6.1. For any fixed z∈T, there exists a unique d∈ℓ∞(Z)such that d(0)=1and dˇ=z∈T. In addition, it is given by d=Uzφ(Uz)*with Uzek=zkek,k∈Z. Proof Substituting dˇ(k)=z in the former conditions, we find the equations zd(2k)=d(2k+1)d(2k)=d(2k−1), whose solution is uniquely determined by d(0). To conclude, just note that Uzφ(Uz)* satisfies the equations thanks to φ(d)(k)=d([k/2]), k∈Z.□ Proposition 6.2. The image of ˇintersects Ton the set of all roots of unity of order 2k, k∈N. Proof Whenever z is a root of unity of order a power 2k the corresponding Uz is in D2, as shown in [1], and dˇ=z with d≐Uzφ(Uz)*. To conclude, we need to prove that nothing else is contained in the intersection. To this aim, note that it is not restrictive to assume d(0)=1 if d satisfies dˇ=z. This allows us to consider d of the form Uzφ(Uz)*. There are two cases to deal with. They can be both worked out with some ideas borrowed from [1]. In the first, z is a root of unity, but not of order a power of 2. This is handled as follows. Since any projection P∈D2 is in the linear algebraic span of {SαSα*}α∈W, we have that Uzφ(Uz)*=∑∣α∣=hcαSαSα*. Without loss of generality, we may suppose that h≥2. By definition, Uzφ(Uz)*e0=e0 so that c(2,…,2)=1. The equality Uzφ(Uz)*e2hl=z2h−1le2hl gives z2h−1l=1, which is absurd as soon as l=1. In the second case, z is not a root of unity. If Uzφ(Uz)* did belong to D2, then there would exist a positive integer k such that ∥Uzφ(Uz)*−∑∣α∣=kcαSαSα*∥<ε with ∑∣α∣=kcαSαSα* being unitary as well. In particular, we would have ∥zie2i−∑∣α∣=kcαSαSα*e2i∥<ε for every integer i. Taking i=0 we would thus get ∥e0−∑∣α∣=kcαSαSα*e0∥=∣1−c(2,…,2)∣<ε, and, in particular, for i=2k−1l ∥z2k−1le2kl−∑∣α∣=kcαSαSα*e2kl∥=∣z2k−1l−c(2,…,2)∣<ε. As z is not a root of unity, then z2k−1 is not a root of unity either. Therefore, {(z2k−1)l}l∈Z is dense in T, which contradicts the inequality above.□ Furthermore, there is a simple invariance property we would like to stress. Lemma 6.3. The image of ˇis invariant under Ad(Uk), for any k∈Z. Proof Set α=Ad(Uk)◦λd◦Ad(U−k). Then the equalities α(U)=UkdˇU−kU and α(x)=x for any x∈D2 are both easily verified. The conclusion is thus arrived at.□ We have finally gathered all the tools necessary to carry out our proof. Theorem 6.4. Let dbe an extendible unitary in D2, then dis localized if and only if dˇis localized. Proof Obviously, all we have to do is prove that d is algebraic if dˇ is. If dˇ is in D2k, then it takes the form dˇ=∑i=02k−1ciUiS2k(S2k)*U−i, where the ci’s are all in T. By the above lemma, the product ∏i=02k−1UidˇU−i=(∏i=02k−1ci)1 is still in the range of the map ˇ. Now Proposition 6.2 tells us that ∏i=02k−1ci is a root of unity of order 2k, for some k∈N. Suppose at first that ∏i=02k−1ci=1. We are, therefore, led to verify that there exists a d˜∈D2k such that d˜Ud˜*U*=dˇ. As above, we can write d˜ as ∑i=02k−1αiUiS2k(S2k)*U−i, which allows to recast the equation for d˜ in terms of its coefficients αi c0=α0α2k−1¯c1=α1α0¯⋮c2k−1=α2k−1α2k−2¯. To begin with, it is not restrictive to suppose α0=1. With this choice, the solution is given by αi=cici−1⋯c1 for every i≥1. Note that the condition ∏i=02k−1ci=1 guarantees the compatibility of the system. Now we can deal with the general case, that is ∏i=02k−1ci=e2πih/2n for some h,n∈N⧹{0}. If we consider d′ˇ≐dˇe−2πih/2n+k, then Proposition 6.2 implies that d′ˇ∈{dUd*U*:d∈U(⋃kD2k)} if and only if dˇ∈{dUd*U*:d∈U(⋃kD2k)}. But now we have that ∏i=02k−1Uid′̌U−i=e−2πih/2n(∏i=02k−1ci)=e−2πih/2ne2πih/2n=1, and the claim follows from the first part of the argument.□ 7. An outlook for further results A natural way to complete the analysis hitherto conducted would require to attempt a generalization of Proposition 5.5 to extend its reach to every d∈U(D2). In this regard, it is worth stressing that if d∈U(D2) is an extendible unitary satisfying dS2=S2, then dˇ must satisfy dˇ=dˇ(0)∏i=1∞(S2*)iUdU*S2i along with the couple of equations {dˇP1=dP1dˇP2r1=(Ud*U*)rφr(d)P2r1,foreveryr≥1 as follows from Theorem 5.1 and an easy induction. Here, we used 2r for 2⋯2 ( r times) to ease the notation. Surprisingly enough, even though dˇ is (over)determined by both the above formulas, the sought extension of Proposition 5.5 has nonetheless proved to be a difficult goal to accomplish. A class of ostensively easier problems, however, arises as soon as further assumptions are made. One could for instance introduce the extra condition dS1k=S1k, for some k≥2, as well as dS2=S2. Still, the problem thus obtained is a hard game to play. Indeed, even dealing with the easiest case, that is k=2, requires a close look at the odometer, see for instance [14], [14, 12, pp. 230, 231]. This is the map T∈Homeo(K) given by (T{xi}i≥1)j≐{1j<k2j=kxjj>k, with k≐inf{j∣xj=1}∈N∪{+∞}, and +∞ is attained only on the constant sequence {222…}. As recalled in Section 2, the map T implements the restriction of Ad(U) to the diagonal D2, that is UdU*(x)=d(Tx) for every x∈{1,2}N≅K. Here is the precise statement of the result alluded to above. Proposition 7.1. The only extendible automorphism λd, d∈U(D2), such that dS2=S2and dS12=S12is the identity. Proof Under our hypotheses, Formula (4.2) at the beginning of Section 4 becomes dˇS1=dS1, which at the spectrum level reads d(1x)=dˇ(1x) for all x∈{1,2}N. Formula (4.1) leads to S2*dˇUdˇU*S2=dˇ, which turns into dˇ(2x)dˇ(1T(x))=dˇ(x) thanks to the equality (S2*dS2)(x)=d(2x). Now the condition dS12=S12 gives dˇ(11x)=d(11x)=1 for all x. Picking x=2i1y we get that dˇ(2i+11y)dˇ(1i+12y)=dˇ(2i1y) for all i≥1, and thus dˇ(2i+11y)=dˇ(2i1y)=dˇ(21y). This says that the sequence {dˇ(2i+11y)} is constant. Therefore, each term of it equals its limit, that is dˇ(22…). Now dˇ(22…)=dˇ(0) (in the canonical representation), hence dˇ(2i1y)=dˇ(0)≐α for every y∈{1,2}N. In particular, we find that dˇ(2y)=α for all y. Under our hypothesis (and by the previous computations) we know that in the canonical representation d(2k+1)=dˇ(2k+1), d(2k)=1, and dˇ(2k)=α for all k∈Z. Thus, if we plug k=1 in (6.1) we get d(1)α=d(1)ď(2)=ď(1)d(2)=d(1)d(2)=d(1), which shows that α=1. The conclusion is now achieved by a straightforward application of Proposition 4.2.□ References 1 V. Aiello , R. Conti and S. Rossi , A look at the inner structure of the 2-adic ring C*-algebra and its automorphism groups, to appear in Publ. RIMS, arXiv preprint:1604.06290, 2016 . doi: 10.4171/PRIMS/54-1-2. 2 R. Conti , J. H. Hong and W. Szymańsky , The restricted Weyl group of the Cuntz algebra and shift endomorphisms , J. Reine Angew. Math. 667 ( 2012 ), 177 – 191 . 3 R. Conti , J. H. Hong and W. Szymański , Endomorphisms of graph algebras , J. Funct. Anal. 263 ( 2012 ), 2529 – 2554 . Google Scholar Crossref Search ADS 4 R. Conti , J. H. Hong and W. Szymański , The Weyl group of the Cuntz algebra , Adv. Math. 231 ( 2012 ), 3147 – 3161 . Google Scholar Crossref Search ADS 5 R. Conti , J. H. Hong and W. Szymański , On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra , Proc. Roy. Soc. Edinburgh 145A ( 2015 ), 269 – 279 . Google Scholar Crossref Search ADS 6 R. Conti , J. Kimberley and W. Szymański , More localized automorphisms of the Cuntz algebras , Proc. Edinburgh Math. Soc. (Series 2) 53 ( 2010 ), 619 – 631 . Google Scholar Crossref Search ADS 7 R. Conti , M. Rørdam and W. Szymański , Endomorphisms of On which preserve the canonical UHF-subalgebra , J. Funct. Anal. 259 ( 2010 ), 602 – 617 . Google Scholar Crossref Search ADS 8 R. Conti , Automorphisms of the UHF algebra that do not extend to the Cuntz algebra , J. Austral. Math. Soc. 89 ( 2010 ), 309 – 315 . Google Scholar Crossref Search ADS 9 R. Conti and W. Szymański , Labeled Trees and Localized Automorphisms of the Cuntz Algebras , Trans. Amer. Math. Soc. 363 ( 2011 ), 5847 – 5870 . Google Scholar Crossref Search ADS 10 J. Cuntz Automorphisms of certain simple C*-algebras, Quantum fields-algebras-processes (Eds. L. Streit ), Springer , Vienna , 1980 . 11 J. Cuntz , Simple C*-algebras generated by isometries , Commun. Math. Phys. 57 ( 1977 ), 173 – 185 . Google Scholar Crossref Search ADS 12 K. R. Davidson , C*-algebras by example Vol. 6. American Math. Soc., Fields Institute Monographs, 1996 . 13 U. Haagerup and K. K. Olesen , Non-inner amenability of the Thompson groups T and V , J. Funct. Anal. 272 ( 2017 ), 4838 – 4852 . Google Scholar Crossref Search ADS 14 D. Kerr and H. Li , Ergodic theory, indipendence and dichotomies, Springer Monographs in Mathematics , Springer International Publishing , Cham , 2016 . 15 N. S. Larsen and X. Li , The 2-adic ring C*-algebra of the integers and its representations , J. Funct. Anal. 262 ( 2012 ), 1392 – 1426 . Google Scholar Crossref Search ADS 16 K. Matsumoto and J. Tomiyama , Outer Automorphisms on Cuntz Algebras , Bull. London Math. Soc. 25 ( 1993 ), 64 – 66 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

Diagonal automorphisms of the 2-adic ring C*-algebra

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Abstract The 2-adic ring C*-algebra Q2 naturally contains a copy of the Cuntz algebra O2 and, a fortiori, also of its diagonal subalgebra D2 with Cantor spectrum. This paper is aimed at studying the group AutD2(Q2) of the automorphisms of Q2 fixing D2 pointwise. It turns out that any such automorphism leaves O2 globally invariant. Furthermore, the subgroup AutD2(Q2) is shown to be maximal abelian in Aut(Q2). Saying exactly what the group is amounts to understanding when an automorphism of O2 that fixes D2 pointwise extends to Q2. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism. 1. Introduction As soon as the Cuntz algebras were introduced in [11], it was quickly realized that studying their endomorphisms and automorphisms would initiate a fruitful research season. The forecast could not possibly be more accurate, for after nearly 40 years they continue to be a major topic and a source of inspiration, as demonstrated by the recent literature that has been accumulating at an impressive rate, see for example [2–9]. Motivated by these works, we found that it is natural to ask ourselves whether such a case study would also provide the right tools to analyze the endomorphisms and automorphisms of other classes of C*-algebras, notably those recently associated with rings, fields and other algebraic objects. In fact, this was one of the main reasons why in [1] we started an investigation of the group Aut(Q2) of unital *-preserving automorphisms of the so-called dyadic ring C*-algebra of the integers Q2, a known C*-algebra (see for example [15] and the references therein) associated to the semidirect product semigroup Z⋊{1,2,22,23,…} that contains a copy of the Cuntz algebra O2 in a canonical way. Inter-alia, it was proved in [1] the useful fact that the canonical diagonal D2 maintains the property of being a maximal abelian subalgebra (MASA) in Q2 also. Actually, more is known, for D2 is even a Cartan subalgebra of both O2 and Q2 [13, Proposition 4.3]. The study there initiated is further developed in the present paper. In particular, our main focus is here on the structure of the set of those automorphisms of Q2 leaving the diagonal MASA D2 pointwise fixed, which will always be denoted by AutD2(Q2). Rather interestingly, this group turns out to be a maximal abelian subgroup of Aut(Q2). Moreover, we show that any of its elements restricts to an automorphism of O2 and it is indeed the unique extension of its restriction. It immediately follows from the analysis carried out in [10] that such restrictions are automorphisms of O2 induced by unitaries in D2, henceforth referred to as diagonal automorphisms for short. The results here obtained lend further support to the idea, already expressed in [1], that the group of automorphisms of Q2 is, in a sense, considerably smaller than that of O2, thus making it reasonable to ask the challenging question whether this group may be computed explicitly up to inner automorphisms. Indeed, we show that any extendible localized diagonal automorphism of O2 is necessarily the product of a gauge automorphism and a localized inner diagonal automorphism. The general case is still out of the reach of the techniques used in this paper instead. Even so, we do spot a necessary and sufficient condition for a diagonal automorphism to extend. Despite all our efforts to exploit the condition, to date this has not aided us in deciding whether the only extendible diagonal automorphisms of O2 are products of gauge and inner diagonal automorphisms. It is quite possible that the answer to this problem will also require to delve further into the fine ergodic properties of the odometer map. 2. Preliminaries and notations We recall the basic definitions and properties of the 2-adic ring C*-algebra so as to make the reading of the present paper suitable for a broader, not necessarily specialized, audience. This is the universal C*-algebra Q2 generated by an isometry S2 and a unitary U such that S2S2*+US2S2*U*=1 and S2U=U2S2. A very informative account of its most relevant properties is given in [15] as well as in our former work [1]. As far as the purposes of this work are concerned, it is important to mention that the Cuntz algebra O2, that is the universal C*-algebra generated by two isometries X1 and X2 such that X1X1*+X2X2*=1 can be thought of as a subalgebra of Q2 via the injective unital ∗-homomorphism that sends X1 to US2 and X2 to S2. As of now this ∗-homomorphism will always be understood without explicit mention, therefore, we simply write O2⊂Q2 to refer to the copy of O2 embedded in Q2 in this way. The 2-adic ring C*-algebra is actually a kind of a more symmetric version of O2, in which the Cuntz isometries S1 and S2 are now intertwined by the unitary U, to wit S2U=US1. This circumstance introduces a higher degree of rigidity, which is ultimately responsible for a shortage of outer automorphisms. Although not completely computed, the group Out(Q2) is nevertheless closely looked over in [1], where it is shown to be considerably smaller than Out(O2), while being still uncountable and non-commutative. To get a better idea of to what extent the former group is smaller than the latter, it is worthwhile to point up that endomorphisms or automorphisms of O2 will not in general extend to Q2, as widely discussed in [1]. Among those that do extend, there are the canonical endomorphism, the gauge automorphisms and the flip-flop. Of these only the first two will play an important role in this work. In particular, the gauge automorphisms will actually play an overriding role, which means they need a bit more exhaustive introduction. On O2, these are the automorphisms αθ acting on each isometry simply multiplying it by eiθ, that is αθ(Si)=eiθSi for i=1,2, where θ is any real number. The action of the one-dimensional torus T on O2 provided by the gauge automorphisms enables us to speak of the gauge invariant subalgebra F2⊂O2, which is by definition the C*-subalgebra whose elements are fixed by all the αθ’s. It is well known that F2 is the UHF algebra of type 2∞. Now the gauge automorphisms are immediately seen to extend to automorphisms αθ˜ of the whole Q2 with αθ˜(U)=U. Less obviously, each αθ˜ is an outer automorphism when it is not trivial, which is proved in [1]. Furthermore, the extended gauge automorphisms allow us to consider the gauge invariant subalgebra of Q2, which we denote by Q2T. Among other things, Q2T is known to be a Bunce–Deddens algebra. It is not particularly hard to prove that Q2T can also be described as the C*-subalgebra of Q2 generated from either F2 or D2 and U, where D2⊂F2 is a notable commutative subalgebra of the Cuntz algebra O2. Commonly referred to as the diagonal subalgebra, D2 is in fact the subalgebra generated by the diagonal projections Pα≐SαSα*, where for any multi-index α=(α1,α2,…,αk)∈⋃n{1,2}n the isometry Sα is the product Sα1Sα2…Sαk. The multi-index notation is rather convenient when making computations in Cuntz algebras and will be used extensively throughout the paper. In particular, we need to recall that ∣α∣ is the length of the multi-index α. By definition, the diagonal D2 is also the inductive limit of the increasing sequence of the finite-dimensional subalgebras D2k⊂D2k+1⊂D2 given by D2k≐span{Pα:∣α∣=k}, k∈N. That D2 is quite a remarkable subalgebra is then seen at the level of its spectrum, for the latter is the Cantor set K, of which many a concrete topological realization is known. However, in what follows we shall always think of it as the Tychonov infinite product {1,2}N. Not of less importance is the canonical endomorphism φ∈End(O2), which is defined on each element x∈O2 as φ(x)=S1xS1*+S2xS2*. By its very definition, it clearly extends to Q2, on which it still acts as a strongly ergodic map, namely ⋂nφn(Q2)=C1, as shown in [1]. The intertwining rules Six=φ(x)Si for any x∈Q2 with i=1,2 still hold true. In addition, the canonical endomorphism preserves the diagonal D2, acting on its spectrum as the usual shift map on {1,2}N. To complete the description of our framework, we still need to single out a distinguished representation of Q2 among the many, namely the so-called canonical representation [15]. This is the representation in which S2 and U are concrete operators acting on the Hilbert space ℓ2(Z) as S2ek=e2k and Uek=ek+1, for every k∈Z, where {ek:k∈Z} is the canonical basis of ℓ2(Z). Finally, in this representation, D2 can be seen as a norm-closed subalgebra of ℓ∞(Z), the diagonal operators with respect to the canonical basis. For any d∈ℓ∞(Z), we denote by d(k) its kth diagonal entries, that is d(k)≐(ek,dek), k∈Z. It is obvious that Ad(U) leaves ℓ∞(Z) invariant. It is slightly less obvious that it also leaves D2 globally invariant. Moreover, the spectrum of D2 is acted on by Ad(U) through the homeomorphism given by the so-called odometer on the Cantor set, which is known to be a uniquely ergodic map, see [12]. Finally, the action of Ad(U) on D2 is compatible with its inductive-limit structure, that is Ad(U)(D2k)=D2k for every k∈N. 3. General structure results on Aut2(2) Given an inclusion of C*-algebras A⊆B, we shall denote by Aut(B,A) the group of those automorphisms of B leaving A globally invariant, and by AutA(B) the group of those fixing A pointwise. Both endomorphisms and automorphism are tacitly assumed to be unital whenever our C*-algebras are unital, as D2 and Q2 are. Having a general content, the following result may possibly be known, cf. [10] for instance. Because we have no explicit reference, a precise statement is nevertheless included along with its proof. To state it as clearly as possible, however, we still need to set some notations. In particular, we recall that if H is any subgroup of a group G, its normalizer is the largest subgroup NH(G) in which H is contained as a normal subgroup. More explicitly, NH(G) can be identified with the set of those g∈G such that gHg−1=H. Finally, if S is any subset of Aut(B), we denote by BS the sub- C*-algebra of B whose elements are fixed by all automorphisms of S. Here follows the result. Proposition 3.1. Let A⊆Bbe a unital inclusion of C*-algebras. Then Aut(B,A)⊆NAutA(B)(Aut(B)); If BAutA(B)=Aone has Aut(B,A)=NAutA(B)(Aut(B)); If Ais a MASA in B, then BAutA(B)=Aand Aut(B,A)=NAutA(B)(Aut(B)). Proof For the first property, let γ∈Aut(B,A) and β∈AutA(B). Since γ◦β◦γ−1(a)=a for all a∈A, we get that γ belongs to NAutA(B)(Aut(B)). For the second, consider γ∈NAutA(B)(Aut(B)) and β∈AutA(B). We have that γ◦β◦γ−1(a)=a for all a∈A, hence β◦γ−1(a)=γ−1(a). This means that γ−1(a)∈BAutA(B) which is by hypothesis equal to A, thus γ∈Aut(B,A). Finally, for the third, we observe that any u∈U(A) gives rise to an element of AutA(B), namely Ad(u). This means that for any x∈BAutA(B) we have uxu*=x. Since A is a MASA, then x∈A and the second property in turn implies that Aut(B,A)=NAutA(B)(Aut(B)).□ By applying the former result to the inclusion D2⊂Q2, the set equality Q2AutD2(Q2)=D2 is immediately got to. Now AutD2(Q2) contains both {αθ˜∣θ∈R} and {Ad(u),u∈U(D2)}, where αθ˜ is the unique extension to Q2 of the gauge automorphism αθ∈Aut(O2), see [1]. Furthermore, the intersection AutD2(Q2)⋂Inn(Q2) is easily seen to reduce to {Ad(u),u∈U(D2)} thanks to maximality of D2 again. Since D2 is globally invariant under Ad(U), for any α∈AutD2(Q2) we have α(UdU*)=UdU*=α(U)dα(U)*foralld∈D2, that is U*α(U) commutes with every d∈U(D2) and therefore by maximality U*α(U)=đα for some đα∈U(D2), which we rewrite as α(U)=Uđα. It is also clear that α−1(U)=Uđα*. To take but one example, when α is an inner automorphism, say Ad(u) for some u in U(D2), the corresponding đAd(u) is nothing but U*uUu*. More importantly, the map α↦đα is easily recognized to be a group homomorphism between AutD2(Q2) and U(D2). Moreover, its kernel coincides with AutQ2T(Q2), cf. Proposition 3.4. Our next goal is to show that the Cuntz–Takesaki unitary uα≐α(S1)S1*+α(S2)S2* belongs to D2 as well. Proposition 3.2. Let αbe in AutD2(Q2). Then the corresponding unitary uα lies in D2. Proof First we observe that α(Si)=uαSi for i=1,2. By maximality of D2⊂Q2, it is enough to prove that uα commutes with the generating projections Pi1i2…ik of D2. This can be easily seen by induction on k, as done by Cuntz for O2. The case of length one reduces to the computation Pi=α(Pi)=α(SiSi*)=uαSiSi*uα*=uαPiuα*. The case of length two entails the computation Pij=α(Pij)=α(SiSjSj*Si*)=uαSi(uαSjSj*uα*)Si*uα*=uαSiSjSj*Si*uα*=uαPijuα*. It is now clear how to go on.□ The following result can be derived at once from the foregoing proposition. In this respect, it is worth recalling the fact, proved in [10], that there exists an explicit group isomorphism between U(D2) and AutD2(O2) given by d↦λd, where λd(Si)=dSi, i=1,2, already showing the abelianness of AutD2(O2). Corollary 3.3. Any α∈AutD2(Q2)restricts to an automorphism of the Cuntz algebra O2fixing pointwise the diagonal D2and it is the unique extension of such restriction. In particular, the group AutD2(Q2)is abelian. Proof It is clear from the previous proposition that α(Si)∈O2 for i=1,2, so that α(O2)⊆O2. Now, uα*Si∈O2 and α(uα*Si)=uα*α(Si)=uα*uαSi=Si, i=1,2, thus showing that indeed α(O2)=O2. We conclude that α is an extension to Q2 of its restriction to O2, and the statement about uniqueness follows at once from the rigidity result, proved in [1, Section 4], that two automorphisms of Q2, coinciding on O2, must be the same. For the last claim, if αi∈AutD2(Q2), i=1,2, we compute α1(α2(U))=α1(Uđα2)=Uđα1đα2=Uđα2đα1=α2(α1(U)) and, similarly, α1(α2(S2))=α1(uα2(S2))=uα2uα1S2=α2(α1(S2)). The conclusion readily follows.□ Every element in AutD2(Q2) can thus be written as the unique extension of an element λd∈AutD2(O2), for some d∈U(D2). We denote such extension as λd˜. However, one should not expect all automorphisms λd with d∈U(D2) to extend to Q2. Denoting by U˜(D2) the set of all d∈U(D2) such that λd is extendible to an automorphism of Q2, it is then easy to deduce from the above discussion that U˜(D2) is actually a group (a subgroup of U(D2)), and there exists a group isomorphism between U˜(D2) and AutD2(Q2) given by d↦λd˜. Thus far we have seen that any α∈AutD2(Q2) acts on U as α(U)=Uđα for some đα∈D2. Now it is also possible to rewrite this relation in the form α(U)=đˇαU, where đˇα is simply given by UđαU* and is still a unitary of D2. As α=λd˜, we can simply write dˇ instead of đˇα=đˇλd˜ for d∈U˜(D2). For the same reason as above, the map d↦dˇ is a group homomorphism from U˜(D2) to U(D2). In fact, this map will turn out to be vital in the next sections. Contrary to what one might expect, though, it has been proved to be a difficult task to establish a priori whether it is norm continuous, possibly because determining its domain U˜(D2) is just another way to recast our main problem. Nevertheless, its kernel can be described quite explicitly. Proposition 3.4. The kernel of the map d↦dˇ (defined on U˜(D2)) is the subgroup of the gauge automorphisms. Actually, one has AutQ2T(Q2)=AutF2(Q2)={αθ˜:θ∈R}. Proof Clearly the condition dˇ=1 is the same as λd˜(U)=U. Since Q2T coincides with C*(U,D2), see for example [1, Section 2], it means that λd˜∈AutQ2T(Q2) so that λd∈AutO2T(O2). The conclusion now readily follows from the fact that the automorphisms of O2 fixing the canonical UHF subalgebra F2=O2T pointwise are precisely the gauge automorphisms, as proved by Cuntz in [10].□ In particular, the restriction map AutD2(Q2)∋λ→λ↾O2∈AutD2(O2) induces a group embedding which allows us to think of the former group as a subgroup of the latter. Therefore, as of now we will simply write AutD2(Q2)⊂AutD2(O2) to mean that. Of course the inclusion is proper. In other words, not all the automorphisms of O2 that leave the diagonal D2 globally invariant will extend. As a matter of fact, very few automorphisms can be extended. Although we do not have a general explicit description of all extendible automorphisms yet, we do have a complete description for a particular class of automorphisms. This is just the subgroup AutD2(O2)loc of those localized automorphisms we mentioned above in passing. Actually, the terminology comes from Quantum Field Theory. Roughly speaking, an automorphism is localized when it preserves the union of the matrix subalgebras. More precisely, an automorphism λu∈Aut(On) is said to be localized when the corresponding unitary u∈U(On) belongs to the algebraic dense subalgebra ⋃kFkn⊂On, where Fkn is generated by the elements of the form SαSβ* with α,β∈{1,…,n}k. Furthermore, the inclusion AutD2(Q2)⊂AutD2(O2) allows us to define a subgroup AutD2(Q2)loc as the intersection AutD2(Q2)⋂Aut(O2)loc. As maintained in the abstract, we will prove that AutD2(Q2)loc is so small that the sole localized automorphisms fixing D2 that extend are the composition of a localized inner automorphism with a gauge automorphism. Before going on with our discussion, we would like to point out a remark for the sake of completeness. Remark 3.5. Let d∈U(D2) and consider the associated automorphism λd of O2. If λd extends to an endomorphism λ of Q2, then λ is actually an automorphism, that is d∈U˜(D2) and λ=λd˜. Indeed, λ(Q2) contains λd(O2)=O2. Moreover, λ(U)=d˜U for a suitable d˜∈U(D2) (the same argument as for automorphisms), so that λ(d˜*U)=d˜*d˜U=U. All in all, the extension is nothing but λd˜ (and d˜=dˇ). Going back to AutD2(Q2), we have shown that it is abelian, but we want to improve our knowledge by proving that it is also maximal abelian in Aut(Q2), in a way that closely resembles what happens for the Cuntz algebra O2 [10]. Here follows the proof. Theorem 3.6. The subgroup AutD2(Q2)is maximal abelian in Aut(Q2). Proof Let α be an automorphism of Q2 that commutes with AutD2(Q2). In particular [α,Ad(u)]=0 for every u∈U(D2), to wit Ad(α(u))=Ad(u). As the center of Q2 is trivial, we see that α(u)=χ(u)u for every u∈U(D2), where χ is a character of the group U(D2). Our result will be proved once we show χ(u)=1 for every u∈U(D2). To this aim, note that the equality α(u)=χ(u)u says that D2 is at least globally invariant under the action of α. With a slight abuse of notation, we still denote by α the restriction of α to D2≅C(K), where K is the Cantor set. Let Φ∈Homeo(K) such that α(f)=f◦Φ for every f∈C(K). The identity obtained above is then recast in terms of Φ as f◦Φ=χ(f)f for every f∈C(K,T). We claim that χ(U(D2))⊂T is at most countable. If so, the theorem can now be easily inferred. Indeed, if Φ is not the identity map, then there exists x∈K such that Φ(x)≠x. Then pick a function f∈C(K,T) such that f(x)=1 and f(Φ(x))=eiθ. The equality f◦Φ=χ(f)f evaluated at x gives χ(f)=eiθ, that is χ is onto T. To really achieve the result we are thus left with the task of proving the claim. This should be quite a standard fact from ergodic theory. However, we do give a complete proof. If μ is any Borel Φ-invariant measure on K, we can consider the Hilbert space L2(K,μ), which is separable because K is metrizable, and the Koopman unitary operator Uϕ associated with Φ, whose action is simply given by UΦ(f)=f◦Φ a.e. for every f∈L2(K). As eigenfunctions of UΦ associated with different eigenvalues are orthogonal and χ(f) is an eigenvalue for every f∈C(K,T), we see that {χ(f):f∈C(K,T)} is a countable set by virtue of separability.□ Remark 3.7. We can also provide an alternative argument for the above result, proving more directly that α(Pβ)=Pβ for all the multi-indices β. First of all we observe that the relation α(d)=χ(d)d implies that the spectrum of the unitary d is invariant under the rotation of χ(d). We begin with the case of P1. Consider the unitary d1=P1+ei2πθP2 with θ≠±1. On the one hand, we know that α(d1)=χ(d1)d1. On the other hand, since the spectrum of d1 is not invariant under non-trivial rotations, we find that χ(d1) must be 1. The same reasoning applies to the unitary d˜1=P1−eiθP2 too, and so we get the equality χ(d˜1)=1, hence α(P1)=α(d1+d˜12)=P1. We now deal with the general case of a Pβ with β being a multi-index of length k in much the same way. Consider the two unitary operators dβ=∑∣γ∣=k,γ≠βeiθPγ+Pβ and d˜β=−∑∣γ∣=k,γ≠βeiθPγ+Pβ. By the same argument as above we still find both α(dβ)=dβ and α(d˜β)=d˜β, and thus α(Pβ)=α(dβ+d˜β2)=Pβ and we are done. 4. Necessary and sufficient conditions for extendability Thanks to the results achieved in the last section, giving a complete non-tautological description of AutD2(Q2) entails studying those unitaries d∈U(D2) for which the corresponding λd∈Aut(O2) may be extended to Q2. This section is mainly concerned with problems of this sort. When an automorphism λd extends, we will say every so often that the corresponding d is extendible itself. This is undoubtedly a slight abuse of terminology, but it aids brevity. Here follows our first result. Lemma 4.1. Let dbe in U(D2). Then λd∈Aut(O2)extends to an endomorphism of Q2if and only if there exists a d˜in U(D2)such that d˜UdS1=dS2d˜U (4.1) d˜UdS2=dS1. (4.2)Moreover, such an extension is automatically an automorphism whenever it exists, that is d∈U˜(D2), and d˜=dˇ. Proof If λd extends, then the two equalities in the statement are easily verified with d˜=dˇ if one applies its extension λd˜ (cf. Remark 3.5) to US1=S2U and US2=S1, respectively, also taking into account that λd˜(U)=dˇU. The converse is dealt with analogously by noting that the pair ( d˜U,dS2) in Q2 still satisfies the defining relations of Q2 and, therefore, by universality, there exists an endomorphism λ of Q2 such that λ(S2)=dS2 and λ(U)=d˜U. But then, λ(S1)=λ(US2)=d˜UdS2=dS1 by Equation (4.2), so that λ extends λd.□ At this point, the reader may be wondering whether it ever happens that d=dˇ. In fact, it turns out that this is never the case unless d=1, namely we have the following result. Proposition 4.2. The unitary d=1is the unique fixed point of the map U˜(D2)∋d↦dˇ∈U(D2). Proof If we work in the canonical representation, we simply need to show that d(k)=1 for every k∈Z. We first handle the even entries. Formula (4.2) becomes dUdS2=dS1, which in turn gives dS2=U*S1=S2. Now by computing the above equality on the vectors of the canonical basis of ℓ2(Z) we get d(2k)=1 for all k∈Z. As for the odd entries, Formula (4.1) leads to dUdS1=dS2dU, which yields d(2k+1)=d(k+1) for all k∈Z. This in turn says all odd entries of d are 1 as well, apart from d(1), which is in fact not determined by this condition. However, it cannot be different from 1, for otherwise d would not even belong to D2.□ Although more focused on the Cuntz algebra O2, the next useful result is included all the same. In fact, we do believe that it may shed some light on applications yet to come. Recall that if d,d′∈U(D2), then Ad(d′)◦λd=λd′dφ(d′)*. In particular, taking d′=d* we get that λd is extendible if and only if λφ(d) is extendible. Proposition 4.3. Let λd∈Aut(O2)be an extendible automorphism. Then either λdis a gauge automorphism or λdUd*U*is outer. Proof First of all we prove that (4.3) To this aim, rewrite Formula (4.2) as dˇUdS2=dˇUdU*S1=dS1. Then, by using the identity =dUd*U* and the multiplicativity of the map ˇ one finds (4.4) whence the claim. Likewise, Formula (4.1) yields dˇUdS1=dˇUdU*S2U=dS2dˇU=dφ(dˇ)S2U and thus dˇUdU*S2=dφ(dˇ)S2, that is dˇd*UdU*φ(dˇ*)S2=S2. (4.5) The former equality actually shows that the automorphism Ad(dˇ*)◦λdUd*U* fixes S2, that is Ad(dˇ*)◦λdUd*U*[S2]=S2. By [16, Corollary B], then either Ad(dˇ*)◦λdUd*U* is the identity or Ad(dˇ*)◦λdUd*U* is an outer automorphism of O2. In the first case dˇφ(dˇ)*=dUd*U*= = hence By applying the last equality to (4.3), we get which proves that dˇ=1, thus λd is a gauge automorphism by Proposition 3.4. In the second case, clearly λdUd*U* is an outer automorphism of O2.□ At any rate, a first application can be given at once. Corollary 4.4. If λd, d∈U(D2)is a non-trivial inner automorphism of O2, then λUdU*is outer. We can now resume to our general discussion. With this in mind, we start by spotting a useful necessary condition on d for the corresponding λd to extend. Proposition 4.5. Let dbe a unitary in D2. If λdextends to Q2, then d(0)=d(−1). Proof Owing to the extendability of λd the isometries dS1=λd(S1) and dS2=λd(S2) are still intertwined, that is they are unitarily equivalent. In particular, their point spectra must coincide. Now the equalities σp(dS1)={d(−1)} and σp(dS2)={d(0)} are both easily checked, hence the thesis follows.□ It goes without saying that the condition is only necessary. Even so, it does have the merit of highlighting a property of which we will have to make an extensive use. Therefore, unless otherwise stated, our unitaries d∈U(D2) will always satisfy the condition d(0)=d(−1). In addition, there is no lack of generality if we further assume that both d(0) and d(−1) equal 1. For if this were not the case, we could always multiply λd by a suitable gauge automorphism, which of course would not affect the extendability of λd, since gauge automorphisms certainly extend to Q2. Finally, we will work in the canonical representation and adopt the Dirac bra-ket notation for rank-one operators: for any given u,v∈ℓ2(Z) the operator w→(v,w)u is denoted by ∣u⟩⟨v∣. This said, we can now state a result which sheds further light on the relation between d and dˇ by relating our setup to some findings in [15]. Theorem 4.6. Let dbe a unitary in D2such that d(0)=d(−1)=1; then there exists the strong limit d∞of the sequence dk≐dφ(d)φ2(d)…φk−1(d)in the canonical representation; λdis weakly inner in the canonical representation restricted to O2; λdextends to a representation of Q2on ℓ2(Z); the automorphism λd∈Aut(O2)extends to an automorphism of Q2if and only if, for a unique α∈T, the strong limit of the sequence xk≐α∣e0⟩⟨e0∣+dφ(d)φ2(d)…φk−1(d)(∑i=0k−1S2iS1S1*(S2*)i)Uφk−1(d*)…φ(d*)d*U*belongs to D2, in which case the limit coincides with dˇand α=dˇ(0); if λdextends to an automorphism λd˜, then λd˜is weakly inner in the canonical representation if ( λdis inner or) dˇ(0)=1. Proof To begin with, we note that for every j∈Z the sequence {dk(j):k∈N}⊂T is eventually constant, as we certainly have dk(j)=d∣j∣(j) for all k≥∣j∣ thanks to φ(d)(k)=d([k/2]), k∈Z and d(0)=d(−1)=1. In particular, this shows that {dk:k∈N} is strongly convergent to a unitary d∞∈ℓ∞(Z). Actually, the operator d∞ thus exhibited implements λd, that is dSi=d∞Sid∞*, i=1,2. Indeed, d∞Sid∞*=limkdkSidk*=limkdkφ(dk)*Si=limkdφk(d*)Si=dSi, (4.6) where we have used the equalities Six=φ(x)Si, for all x∈O2 and i=1,2 and that φk(d) strongly converges to 1. Because Ad(d∞) restricts to O2 as λd, it also restricts to Q2 if only as a representation of the latter algebra on the Hilbert space ℓ2(Z). To ease some of the computations we need to make, it is now particularly convenient to introduce the sequence of projections Qk≐∣e0⟩⟨e0∣+∑i=0k−1S2iS1S1*(S2*)i, which act on ℓ2(Z). It is then not difficult to see that now the xk’s in the statement take on the much simpler form xk=(α−1)∣e0⟩⟨e0∣+QkdkUdk*U*. Now Qk strongly converges to the identity I, as shown by straightforward computations. Furthermore, the sequence Udk*U* converges too, since dk does. This shows that xk is strongly convergent to a limit δ≐(α−1)∣e0⟩⟨e0∣+d∞Ud∞*U*. Note that δ is a unitary lying in ℓ∞(Z)=D2″, for the equality see [1, Section 2]. We also point out the equality dk=dφ(dk−1), k∈{2,3,…}∪{∞}, which is necessary to carry out some of the following computations. We can now deal with 4. We start with the if part. In view of Lemma 4.1, it is enough to show that S2˜U˜≐U˜2S2˜ and S1˜≐U˜S2˜, where U˜=δU∈Q2 and Si˜=dSi. As for the first equality, we rewrite δU as δU=(α−1)∣e0⟩⟨e−1∣+d∞Ud∞*, where ∣e0⟩⟨e−1∣(v)=v(−1)e0 for all v∈ℓ2(Z). If we now use the expression obtained above, we can compute U˜2S2˜ as δUδUdS2=((α−1)∣e0⟩⟨e−1∣+d∞Ud∞*)2dS2=(α−1)∣e0⟩⟨e−1∣d∞Ud∞*dS2+d∞U2d∞*dS2=(α−1)∣e0⟩⟨e−1∣+d∞U2d∞*dS2, since ∣e0⟩⟨e−1∣d∞Ud∞*dS2ek is always zero unless k=−1, in which case it is equal to e0, while dS2δU=dS2((α−1)∣e0⟩⟨e−1∣+d∞Ud∞*)=(α−1)∣e0⟩⟨e−1∣+dS2d∞Ud∞*, since dS2e0=e0. It remains to show that d∞U2d∞*dS2=dS2d∞Ud∞*. However, these two terms are equal to d∞U2S2d∞* and d∞S2Ud∞*, respectively, where we used equation (4.6). The second equality is dealt with by means of a still easier computation: δUdS2=((α−1)∣e0⟩⟨e−1∣+d∞Ud∞*)dS2=d∞Ud∞*dS2=d∞S1d∞*=dS1. It is now clear that dˇ=(α−1)∣e0⟩⟨e0∣+d∞Ud∞*U*∈Q2 and it follows at once that dˇ(0)=α. For the only if part, if we think of λd as a representation of O2 on ℓ2(Z), the conclusion follows by the uniqueness pointed out in [15, Remark 4.2] (applied to dˇU). Finally, the condition dˇ(1)=1 leads to the equality dˇ=d∞Ud∞*U*, whence λd˜(U)=dˇU=d∞Ud∞*=Ad(d∞)(U), which says λd˜ is implemented by d∞.□ Needless to say, if λd is inner then (it extends to Q2 and) its extension in still inner. Conversely, if λd extends to an inner automorphism of Q2, then λd itself must be inner by the maximality of D2 in Q2 [1, Section 3.1]. Note that if λd=Ad(d′), with d′∈U(D2) then dˇ(0)=d′(0)d′(−1)¯. If moreover dˇ(0)=1, then d′ is nothing but d∞, which is thus in D2. Finally, it is not difficult to realize that the above theorem could also be set in other representations. Nevertheless, we shall refrain from discussing this issue any further, not least because it goes beyond the scope of the present work. For the results in the next section, we need to take a closer look at inner diagonal automorphisms, to which the rest of this section is addressed. To do that, we first prove a general lemma. Lemma 4.7. For any d∈U(D2)the sequence (S2*)ndS2nconverges in norm to the scalar d(0)1. Proof Given any ε>0 we can pick an algebraic dk∈D2k such that ∥d−dk∥<ε2. As ∥(S2*)ndS2n−d(0)1∥≤∥(S2*)n(d−dk)S2n∥+∥(S2*)ndkS2n−d(0)1∥≤ε2+∥(S2*)ndkS2n−d(0)1∥. So the conclusion is obtained if we can also prove that ∥(S2*)ndkS2n−d(0)1∥ may be made as small as wished. To this aim, it is enough to write dk as a finite linear combination of projections, say dk=∑∣α∣=kdαPα, where all the coefficients dα are in T. Thought of as a continuous function on the Cantor set, the unitary dk is nothing but dk(x)=∑∣α∣=kdαχα(x), and so there holds the inequality supx∈{1,2}N∣d(x)−∑∣α∣=kdαχα(x)∣<ε2. In particular, we also have ∣d({222…})−d22…2∣=∣d(0)−d22…2∣<ε2. But as soon as n is greater than k, the product (S2*)ndkS2n reduces to d22…21, and the conclusion is thus proved, being more exactly ∥(S2*)ndS2n−d(0)1∥≤ε.□ Remark 4.8. The same conclusion is also got to in the canonical representation. Indeed, since (S2*)ndS2nek=d(k2n)ek for any k∈Z, the claim amounts to proving that limnd(k2n)=d(0) for any d∈D2, with the limit being uniform in k. This is again easily proved by approximating in norm any such d with algebraic unitaries as closely as necessary, which is much the same as in the proof above. In our next result, there appear infinite products in a C*-algebraic framework. Since this is a topic seldom discussed in the literature, some comments as to which sense the convergence is understood in are necessary to dispel any possible doubt. The most refined notion one could work with essentially dates back to von Neumann, and is the one intended with respect to the direct net of finite subsets of N ordered by inclusion. However, we shall not need to be that demanding. For our purposes, we may as well make do with the usual notion that an infinite product ∏ai converges (in norm) if the sequence pn≐∏i=1nai does. Proposition 4.9. Let dbe a unitary in D2such that d(−1)=d(0)=1. Consider the following claims: The automorphism λdis inner. The automorphism λdextends to Q2and the infinite product ∏i=1∞(S2*)idS2iconverges in norm to a unitary in D2. There exists a unitary d′∈D2such that λd(S2)=d′S2d′*.Then we have the chain of implications 1⇒2⇔3. Moreover, when λdis inner the above infinite product converges to d′*, with d′being the unique unitary in D2such that d′(0)=1and d′φ(d′)*=das well as satisfying λd(S2)=d′S2d′*. Proof For the implication 1⇒2, we only need to prove that the infinite product converges. To this aim, let d′∈U(D2) such that d′(0)=1 and d′φ(d′)*=d. By means of a simple computation by induction, we gain the formula ∏i=1n(S2*)idS2i=(S2*)nd′S2nd′*. So the conclusion is arrived at by a straightforward application of the former lemma. For 2⇒3, we begin observing that it can be easily seen that ∏j=1n(S2*)jdS2j=(S2*)ndnS2n. By the following computations, we get (∏j=1n(S2*)jd*S2j)S2(∏j=1n(S2*)jdS2j)=((S2*)ndn*S2n)S2((S2*)ndnS2n)=(S2*)ndn*S2[S2n(S2*)n]dnS2n=(S2*)ndn*S2dnS2n=(S2*)ndn*φ(dn)S2S2n=(S2*)nd*φn(d)S2n+1=((S2*)nd*S2n)dS2. Therefore, it follows that limn(∏j=1n(S2*)jd*S2j)S2(∏j=1n(S2*)jdS2j)=limnd(S2*)nd*S2nS2=d(limn(S2*)nd*S2n)S2=dS2. So if we denote by d′ the limit of ∏j=1n(S2*)jd*S2j, we have the equality d′S2d′*=dS2. Finally the implication 3⇒2 is dealt with by similar computations as in the proof of 1⇒2.□ A well-known sufficient condition for the product in point 2 to exist (even with respect to the von Neumann notion of convergence) is that ∑i=1∞∥1−(S2*)idS2i∥<∞. 5. Localized extendible automorphisms While chiefly devoted to presenting our solution to the extension problem for localized automorphisms, this section also includes some results on general, that is possibly non-localized, unitaries. In fact, findings of this sort are interesting in their own right as well as being necessary tools to attack the case of localized automorphisms. First, we prove a structural result about the diagonal elements with the extension property. Theorem 5.1. If d∈U(D2)is an extendible unitary, then d, dˇand Usatisfy the following identity: d=UdU*dˇ(S1S1*+φ(dˇ)*S2S2*). (5.1) Proof The claim follows as in the proof of Proposition 4.3. Indeed, dS1=dˇUdU*S1dS2=dˇφ(dˇ*)UdU*S2 and we are done.□ Now a handful of corollaries can be easily derived. Corollary 5.2. Let λd, d∈U(D2), be an extendible automorphism. Then dˇ=dˇ(0)∏i=1∞(S2*)id*UdU*S2i. Proof By Formula (5.1), we have that ∏i=1k(S2*)idUd*U*S2i=∏i=1k(S2*)i(ď(S1S1*+φ(ď)*S2S2*))S2i=∏i=1k(S2*)i(ďφ(ď)*S2S2*)S2i=∏i=1k(S2*)i(ďφ(ď)*)S2i=(S2*)k(ďφ(ď)*)kS2k=(S2*)kďφk(ď)*S2k=(S2*)kďS2kď*, which converges in norm to dˇ*dˇ(0) as k→+∞ by Lemma 4.7.□ Corollary 5.3. Let λd, d∈U(D2), be an extendible automorphism. Then the product ∏i=1∞(S2*)idS2isits in D2if and only if ∏i=1∞(S2*)iUdU*S2idoes. Proof By formula (5.1) we have that ∏i=1k(S2*)idS2i=∏i=1k(S2*)i[dˇUdU*(S1S1*+φ(dˇ)*S2S2*)]S2i=∏i=1k(S2*)i[dˇφ(dˇ)*UdU*S2S2*]S2i=∏i=1k(S2*)i[dˇφ(dˇ)*UdU*]S2i=(∏i=1k(S2*)idˇφ(dˇ)*S2i)(∏i=1k(S2*)iUdU*S2i). Since the sequence ∏i=1k(S2*)idˇφ(dˇ)*S2i is norm-convergent as k→+∞ we have the claim.□ The former corollary applies in particular to localized unitaries. But in this case, only finitely many terms occur in the infinite product that gives the corresponding dˇ. More precisely, if d∈U(D2k) then dˇ=dˇ(0)∏i=1k−1(S2*)id*UdU*S2i. As a result, the following corollary is proved forthwith. Corollary 5.4. Let kbe a non-negative integer and let d∈U(D2k)be such that λdis extendible. Then dˇ∈U(D2k−1). This simple yet effective remark enables us to prove a preliminary result. Proposition 5.5. Let b∈U(D2k)be such that λbis extendible and bS2=S2. Then b=1. Proof We proceed by induction. The case k=1 follows immediately from Proposition 4.5. Now let us suppose that b∈U(D2k) with k>1. Notice that one must have b=bS1S1*+S2S2* and, moreover, bS1=λb(S1)=λb(US2)=bˇUS2=bˇS1, thus giving b=bˇS1S1*+S2S2*. By Corollary 5.4, the unitary bˇ, and hence b, is in U(D2k−1), and we are done.□ We are now in a position to prove the main result of the section. Theorem 5.6. Let d∈U(D2k)be such that λdis extendible. Then λdis the product of a gauge automorphism and an inner localized automorphism. Proof As explained, we can suppose without loss of generality that d(0)=d(−1)=1. Proposition 4.9 yields a d′ in U(D2k−1) such that Ad(d′*)◦λd(S2)=S2. The conclusion readily follows by Proposition 5.5 applied to b=d′*φ(d′)d∈U(D2k).□ At this point, it is worthwhile to stress that the proof of the main theorem also shows that for localized automorphisms the three conditions in Proposition 4.9 are actually equivalent. Indeed, whenever d is localized the infinite product in 2. does converge being a finite product, and thus the unitary in 3. is localized too. There is yet another consequence of Proposition 5.5 that deserves to be highlighted as a separate statement. Corollary 5.7. The groups AutD2(Q2)locand AutC*(S2)(Q2)have trivial intersection. One might wonder whether the subscript loc may be got rid of altogether in the last statement. Remark 5.8. The above discussion implies that if dn,d∈U(D2) are such that λdn and λd are extendible to Q2 and ∥dn−d∥→0 as n→∞, then one has the continuity property ∥dnˇ(S1S1*+φ(dnˇ)*S2S2*)−dˇ(S1S1*+φ(dˇ)*S2S2*)∥→0 as n→∞. If we multiply by S1S1*, we see that ∥dˇnS1−dˇS1∥ goes to zero as well. In order to prove the continuity of the map d→dˇ with respect to the norm topology, we should also check limn∥dˇnS2−dˇS2∥=0. As a matter of fact, all we can prove is the convergence only holds strongly. This is indeed a consequence of a general fact applied to {dˇ*dnˇ}: for any sequence {un}⊂U(D2) such that both limn∥unS1−S1∥ and limn∥unS2un*−S2∥ are zero, one also has limn∣un(2k)−1∣=0, for every k∈Z. However, the limit will in general fail to be uniform in k.This is because we can only say ∣un(2k)−1∣ is less than the sum ∣un(2k)−un(k)∣+∣un(k)−un(k2)∣+…+∣un(k2h)−1∣, with h being ≐min{l:k2lisodd}, and so the number of terms occurring in the sum depends upon what k is (at worst, when k=2l, exactly l terms are needed). 6. Localizing dˇ There is a last property of the group homomorphism ˇ that is well worth a thorough discussion. We have already observed that dˇ is localized whenever d is. Moreover, its kernel is obviously made up of localized unitaries. This seems to indicate that the other way around might also hold true, namely that an extendible d∈U(D2), whose dˇ is localized, has to be localized itself. This is quite the situation we are in. Far from immediate, the proof requires some preliminary work to do instead. More precisely, we need to give a closer look at the image of our homomorphism. To do so, we point out that between the entries of d and dˇ with respect to the canonical basis {ek}k∈Z in the canonical representation on ℓ2(Z) there hold some relations. More precisely, by evaluating (4.1) and (4.2) on ek−1 and ek, respectively, we get the following couple of equations: d(2k−1)dˇ(2k)=dˇ(k)d(2k) (6.1) dˇ(2k+1)d(2k)=d(2k+1) (6.2) to be satisfied by every integer k. Furthermore, the former equation can be recast in terms of the canonical endomorphism φ as dˇ*φ(dˇ)S2S2*=d*UdU*S2S2*. Here below follows a technical result. Lemma 6.1. For any fixed z∈T, there exists a unique d∈ℓ∞(Z)such that d(0)=1and dˇ=z∈T. In addition, it is given by d=Uzφ(Uz)*with Uzek=zkek,k∈Z. Proof Substituting dˇ(k)=z in the former conditions, we find the equations zd(2k)=d(2k+1)d(2k)=d(2k−1), whose solution is uniquely determined by d(0). To conclude, just note that Uzφ(Uz)* satisfies the equations thanks to φ(d)(k)=d([k/2]), k∈Z.□ Proposition 6.2. The image of ˇintersects Ton the set of all roots of unity of order 2k, k∈N. Proof Whenever z is a root of unity of order a power 2k the corresponding Uz is in D2, as shown in [1], and dˇ=z with d≐Uzφ(Uz)*. To conclude, we need to prove that nothing else is contained in the intersection. To this aim, note that it is not restrictive to assume d(0)=1 if d satisfies dˇ=z. This allows us to consider d of the form Uzφ(Uz)*. There are two cases to deal with. They can be both worked out with some ideas borrowed from [1]. In the first, z is a root of unity, but not of order a power of 2. This is handled as follows. Since any projection P∈D2 is in the linear algebraic span of {SαSα*}α∈W, we have that Uzφ(Uz)*=∑∣α∣=hcαSαSα*. Without loss of generality, we may suppose that h≥2. By definition, Uzφ(Uz)*e0=e0 so that c(2,…,2)=1. The equality Uzφ(Uz)*e2hl=z2h−1le2hl gives z2h−1l=1, which is absurd as soon as l=1. In the second case, z is not a root of unity. If Uzφ(Uz)* did belong to D2, then there would exist a positive integer k such that ∥Uzφ(Uz)*−∑∣α∣=kcαSαSα*∥<ε with ∑∣α∣=kcαSαSα* being unitary as well. In particular, we would have ∥zie2i−∑∣α∣=kcαSαSα*e2i∥<ε for every integer i. Taking i=0 we would thus get ∥e0−∑∣α∣=kcαSαSα*e0∥=∣1−c(2,…,2)∣<ε, and, in particular, for i=2k−1l ∥z2k−1le2kl−∑∣α∣=kcαSαSα*e2kl∥=∣z2k−1l−c(2,…,2)∣<ε. As z is not a root of unity, then z2k−1 is not a root of unity either. Therefore, {(z2k−1)l}l∈Z is dense in T, which contradicts the inequality above.□ Furthermore, there is a simple invariance property we would like to stress. Lemma 6.3. The image of ˇis invariant under Ad(Uk), for any k∈Z. Proof Set α=Ad(Uk)◦λd◦Ad(U−k). Then the equalities α(U)=UkdˇU−kU and α(x)=x for any x∈D2 are both easily verified. The conclusion is thus arrived at.□ We have finally gathered all the tools necessary to carry out our proof. Theorem 6.4. Let dbe an extendible unitary in D2, then dis localized if and only if dˇis localized. Proof Obviously, all we have to do is prove that d is algebraic if dˇ is. If dˇ is in D2k, then it takes the form dˇ=∑i=02k−1ciUiS2k(S2k)*U−i, where the ci’s are all in T. By the above lemma, the product ∏i=02k−1UidˇU−i=(∏i=02k−1ci)1 is still in the range of the map ˇ. Now Proposition 6.2 tells us that ∏i=02k−1ci is a root of unity of order 2k, for some k∈N. Suppose at first that ∏i=02k−1ci=1. We are, therefore, led to verify that there exists a d˜∈D2k such that d˜Ud˜*U*=dˇ. As above, we can write d˜ as ∑i=02k−1αiUiS2k(S2k)*U−i, which allows to recast the equation for d˜ in terms of its coefficients αi c0=α0α2k−1¯c1=α1α0¯⋮c2k−1=α2k−1α2k−2¯. To begin with, it is not restrictive to suppose α0=1. With this choice, the solution is given by αi=cici−1⋯c1 for every i≥1. Note that the condition ∏i=02k−1ci=1 guarantees the compatibility of the system. Now we can deal with the general case, that is ∏i=02k−1ci=e2πih/2n for some h,n∈N⧹{0}. If we consider d′ˇ≐dˇe−2πih/2n+k, then Proposition 6.2 implies that d′ˇ∈{dUd*U*:d∈U(⋃kD2k)} if and only if dˇ∈{dUd*U*:d∈U(⋃kD2k)}. But now we have that ∏i=02k−1Uid′̌U−i=e−2πih/2n(∏i=02k−1ci)=e−2πih/2ne2πih/2n=1, and the claim follows from the first part of the argument.□ 7. An outlook for further results A natural way to complete the analysis hitherto conducted would require to attempt a generalization of Proposition 5.5 to extend its reach to every d∈U(D2). In this regard, it is worth stressing that if d∈U(D2) is an extendible unitary satisfying dS2=S2, then dˇ must satisfy dˇ=dˇ(0)∏i=1∞(S2*)iUdU*S2i along with the couple of equations {dˇP1=dP1dˇP2r1=(Ud*U*)rφr(d)P2r1,foreveryr≥1 as follows from Theorem 5.1 and an easy induction. Here, we used 2r for 2⋯2 ( r times) to ease the notation. Surprisingly enough, even though dˇ is (over)determined by both the above formulas, the sought extension of Proposition 5.5 has nonetheless proved to be a difficult goal to accomplish. A class of ostensively easier problems, however, arises as soon as further assumptions are made. One could for instance introduce the extra condition dS1k=S1k, for some k≥2, as well as dS2=S2. Still, the problem thus obtained is a hard game to play. Indeed, even dealing with the easiest case, that is k=2, requires a close look at the odometer, see for instance [14], [14, 12, pp. 230, 231]. This is the map T∈Homeo(K) given by (T{xi}i≥1)j≐{1j<k2j=kxjj>k, with k≐inf{j∣xj=1}∈N∪{+∞}, and +∞ is attained only on the constant sequence {222…}. As recalled in Section 2, the map T implements the restriction of Ad(U) to the diagonal D2, that is UdU*(x)=d(Tx) for every x∈{1,2}N≅K. Here is the precise statement of the result alluded to above. Proposition 7.1. The only extendible automorphism λd, d∈U(D2), such that dS2=S2and dS12=S12is the identity. Proof Under our hypotheses, Formula (4.2) at the beginning of Section 4 becomes dˇS1=dS1, which at the spectrum level reads d(1x)=dˇ(1x) for all x∈{1,2}N. Formula (4.1) leads to S2*dˇUdˇU*S2=dˇ, which turns into dˇ(2x)dˇ(1T(x))=dˇ(x) thanks to the equality (S2*dS2)(x)=d(2x). Now the condition dS12=S12 gives dˇ(11x)=d(11x)=1 for all x. Picking x=2i1y we get that dˇ(2i+11y)dˇ(1i+12y)=dˇ(2i1y) for all i≥1, and thus dˇ(2i+11y)=dˇ(2i1y)=dˇ(21y). This says that the sequence {dˇ(2i+11y)} is constant. Therefore, each term of it equals its limit, that is dˇ(22…). Now dˇ(22…)=dˇ(0) (in the canonical representation), hence dˇ(2i1y)=dˇ(0)≐α for every y∈{1,2}N. In particular, we find that dˇ(2y)=α for all y. Under our hypothesis (and by the previous computations) we know that in the canonical representation d(2k+1)=dˇ(2k+1), d(2k)=1, and dˇ(2k)=α for all k∈Z. Thus, if we plug k=1 in (6.1) we get d(1)α=d(1)ď(2)=ď(1)d(2)=d(1)d(2)=d(1), which shows that α=1. The conclusion is now achieved by a straightforward application of Proposition 4.2.□ References 1 V. Aiello , R. Conti and S. Rossi , A look at the inner structure of the 2-adic ring C*-algebra and its automorphism groups, to appear in Publ. RIMS, arXiv preprint:1604.06290, 2016 . doi: 10.4171/PRIMS/54-1-2. 2 R. Conti , J. H. Hong and W. Szymańsky , The restricted Weyl group of the Cuntz algebra and shift endomorphisms , J. Reine Angew. Math. 667 ( 2012 ), 177 – 191 . 3 R. Conti , J. H. Hong and W. Szymański , Endomorphisms of graph algebras , J. Funct. Anal. 263 ( 2012 ), 2529 – 2554 . Google Scholar Crossref Search ADS 4 R. Conti , J. H. Hong and W. Szymański , The Weyl group of the Cuntz algebra , Adv. Math. 231 ( 2012 ), 3147 – 3161 . Google Scholar Crossref Search ADS 5 R. Conti , J. H. Hong and W. Szymański , On conjugacy of maximal abelian subalgebras and the outer automorphism group of the Cuntz algebra , Proc. Roy. Soc. Edinburgh 145A ( 2015 ), 269 – 279 . Google Scholar Crossref Search ADS 6 R. Conti , J. Kimberley and W. Szymański , More localized automorphisms of the Cuntz algebras , Proc. Edinburgh Math. Soc. (Series 2) 53 ( 2010 ), 619 – 631 . Google Scholar Crossref Search ADS 7 R. Conti , M. Rørdam and W. Szymański , Endomorphisms of On which preserve the canonical UHF-subalgebra , J. Funct. Anal. 259 ( 2010 ), 602 – 617 . Google Scholar Crossref Search ADS 8 R. Conti , Automorphisms of the UHF algebra that do not extend to the Cuntz algebra , J. Austral. Math. Soc. 89 ( 2010 ), 309 – 315 . Google Scholar Crossref Search ADS 9 R. Conti and W. Szymański , Labeled Trees and Localized Automorphisms of the Cuntz Algebras , Trans. Amer. Math. Soc. 363 ( 2011 ), 5847 – 5870 . Google Scholar Crossref Search ADS 10 J. Cuntz Automorphisms of certain simple C*-algebras, Quantum fields-algebras-processes (Eds. L. Streit ), Springer , Vienna , 1980 . 11 J. Cuntz , Simple C*-algebras generated by isometries , Commun. Math. Phys. 57 ( 1977 ), 173 – 185 . Google Scholar Crossref Search ADS 12 K. R. Davidson , C*-algebras by example Vol. 6. American Math. Soc., Fields Institute Monographs, 1996 . 13 U. Haagerup and K. K. Olesen , Non-inner amenability of the Thompson groups T and V , J. Funct. Anal. 272 ( 2017 ), 4838 – 4852 . Google Scholar Crossref Search ADS 14 D. Kerr and H. Li , Ergodic theory, indipendence and dichotomies, Springer Monographs in Mathematics , Springer International Publishing , Cham , 2016 . 15 N. S. Larsen and X. Li , The 2-adic ring C*-algebra of the integers and its representations , J. Funct. Anal. 262 ( 2012 ), 1392 – 1426 . Google Scholar Crossref Search ADS 16 K. Matsumoto and J. Tomiyama , Outer Automorphisms on Cuntz Algebras , Bull. London Math. Soc. 25 ( 1993 ), 64 – 66 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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The Quarterly Journal of MathematicsOxford University Press

Published: Sep 1, 2018

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