Metric semigroups that determine locally compact groups

Metric semigroups that determine locally compact groups Abstract Let G be a locally compact group. Let A be any one of the (complex) Banach algebras: L1(G), M(G), WAP(G) and LUC(G), consisting of integrable functions, regular Borel complex measures, weakly almost periodic functions and bounded left uniformly continuous functions, respectively, on G. We show that the metric semigroup A+1:={f∈A:f≥0and∥f∥=1} (the convex structure is not considered) is a complete invariant for G. 1. Introduction In this paper, we find several new and simple complete invariants for locally compact groups. Let G and H be locally compact groups. Wendel showed in [24] (respectively, Johnson showed in [10]) that G and H are isomorphic if and only if there exists an isometric algebra isomorphism Φ:L1(G)→L1(H) (respectively, Φ:M(G)→M(H)). Optimistically, as Φ(sf)=sΦ(f), information in the one dimensional subspace {sf:s∈C} is somehow encoded in the element {f}. This leads to a quest of a ‘smaller invariant’. As a candidate, however, the unit sphere of L1(G) is not closed under the convolution product, and hence cannot be served as an invariant for G. On the other hand, Kawada showed in [11] that G and H are isomorphic whenever there is an algebra isomorphism Ψ:L1(G)→L1(H) satisfying: Ψ(f)≥0 if and only if f≥0. Observe that L1(G)+1, the positive part of the unit sphere of L1(G), is closed under the convolution product. This suggests us to consider L1(G)+1 as a candidate of a complete invariant of G. In this article, we will show that the metric and the semigroup structures of L1(G)+1, or those of M(G)+1 (note that the convexity is not needed), determine G. This result supplements the above-mentioned results of Wendel [24], Johnson [10] and Kawada [11]. Furthermore, Ghahramani et al. [8], as well as Lau and McKennon [13], showed that either one of the dual Banach algebras LUC(G)* and WAP(G)* determines G, too. We will also show that the positive parts of the unit spheres of LUC(G)* and WAP(G)* are complete invariants for G. For a subset S⊆E of an ordered Banach space E, we set S+1≔{f∈S:∥f∥=1;f≥0}. Our main results (namely, Theorems 2.4 and 2.5) can be subsumed and simplified in the following statement. Theorem 1.1. Two locally compact groups Gand Hare isomorphic as topological groups if and only if any one of the following holds: L1(G)+1≅L1(H)+1as metric semigroups; M(G)+1≅M(H)+1as metric semigroups; (WAP(G)*)+1≅(WAP(H)*)+1as metric semigroups; (LUC(G)*)+1≅(LUC(H)*)+1as metric semigroups. We will obtain the above assertions by verifying that those metric preserving semigroup isomorphisms actually extend to isometric algebra isomorphisms between the whole Banach algebras, and then the corresponding established results in [10, 11, 24] apply. This task is non-trivial. Although it has been shown in [19] that metric preserving bijection from the unit sphere of L1(G;R) (the space of real valued integrable functions) onto that of L1(H;R) extends to a real linear isometry from L1(G;R) onto L1(H;R), neither this statement nor the argument in [19] can be used in our cases. In fact, on top of elementary arguments, our proofs also depend on a theorem of Dye from [6] and its applications given in [16], which are results concerning W*-algebras. 2. The proof of the main theorem Theorem 1.1 is a consequence of the following result, which should be of independent interest (in particular, it tells us that the metric structure on the normal state space of a W*-algebra encodes its convex structure, when the algebra is abelian). Proposition 2.1. Let Mand Nbe W*-algebras, with one of them being abelian. If Φ:(M*)+1→(N*)+1is a bijection satisfying ∥Φ(f)−Φ(g)∥=∥f−g∥(f,g∈M), then there is a *-isomorphism Θ:M→Nsatisfying Θ*−1∣(M*)+1=Φ. Let us first do some preparation for the proof of Proposition 2.1. In the following, for any subset Δ of a set X, we denote by χΔ:X→{0,1} the characteristic function of Δ. Moreover, for any function g:X→C, we set supp g to be the support of g; namely, suppg≔{x∈X:g(x)≠0}. Lemma 2.2. Suppose that (X,Ω,μ)is a measure space and n∈N. (a) Let E∈Ωand c>0such that 0<cμ(E)≤1. Suppose that f∈L1(μ)+with suppf⊆Esuch that ∫Xfdμ=cμ(E). Then f=cχEif and only if for any Δ∈Ωwith Δ⊆Eand μ(Δ)>0, there exists gΔ∈L1(μ)+1satisfying suppgΔ⊆Δand ∥f−gΔ∥L1(μ)=1+cμ(E)−2cμ(Δ). (b) Let E1,…,En∈Ωand c1,…,cn>0satisfying ∑k=1nckμ(Ek)=1as well as Ei∩Ej=∅and μ(Ei)>0for any 1≤i≠j≤n. Consider f∈L1(μ)+1with ∫Elfdμ=clμ(El)(l=1,…,n).Then f=∑k=1nckχEkif and only if for any l∈{1,…,n}and Δ∈Ωwith Δ⊆Eland μ(Δ)>0, there exists hΔ,l∈L1(μ)+1satisfying supphΔ,l⊆Δand ∥f−hΔ,l∥L1(μ)=2−2clμ(Δ). Proof (a) ⇒). This implication is clear if we take gΔ≔1μ(Δ)χΔ. ⇐). For any r>0, we set Δr≔{x∈E:f(x)≤r}. Assume on the contrary that f≠cχE. Then one can find d∈(0,c) with μ(Δd)>0 (otherwise, f(x)≥c for μ-almost every x∈E, which, together with ∫Xfdμ=cμ(E), will imply f=cχE). Hence, we can find e∈(0,d] satisfying ∫Δdfdμ=eμ(Δd). Suppose gΔd∈L1(μ)+1 is as in the statement. Since suppgΔd⊆Δd, we know that ∥f−gΔd∥L1(μ)=∥f·χE⧹Δd∥L1(μ)+∥gΔd−f·χΔd∥L1(μ)≥∫E⧹Δdfdμ+(1−eμ(Δd))=1+cμ(E)−2eμ(Δd). This, together with the hypothesis, tells us that 2cμ(Δd)≤2eμ(Δd), and this contradicts with e<c. (b) ⇒). This implication is clear if we set hΔ,l≔1μ(Δ)χΔ. ⇐). Fix any l∈{1,…,n} and set fl≔χEl·f. Let Δ∈Ω with Δ⊆El and μ(Δ)>0. Consider hΔ,l∈L1(μ)+1 to be the element as in the statement. The equality ∥f−hΔ,l∥L1(μ)=∥f−fl∥L1(μ)+∥fl−hΔ,l∥L1(μ)=(1−clμ(El))+∥fl−hΔ,l∥L1(μ) implies ∥fl−hΔ,l∥L1(μ)=1+clμ(El)−2clμ(Δ). Thus, we conclude from part (a) that fl=clχEl. Since ∑k=1nckμ(Ek)=1 and ∫Xfdμ=1, we know that f=∑k=1nfk.□ Lemma 2.3. Let (X,Ω,μ)be a semi-finite measure space. If Λ:L1(μ)+1→L1(μ)+1is a bijection satisfying ∥Λ(f)−Λ(g)∥=∥f−g∥andμ(suppg⧹suppΛ(g))=0(f,g∈L1(μ)+1), (2.1)then Λis the identity map. Proof Since the set of positive simple functions with norm one is dense in L1(μ)+1, we only need to show that Λ(∑k=1nckχEk)=∑k=1nckχEk for any positive scalars c1,…,cn, and disjoint subsets E1,…,En∈Ω with μ(Ei)>0 ( i=1,…,n) satisfying ∑k=1nckμ(Ek)=1. Let us set f≔∑k=1nckχEk and fix an arbitrary integer l∈{1,…,n}. Let us also denote Λ(f)l≔Λ(f)·χElandg≔χEl/μ(El). Then ∥f−g∥L1(μ)=2−2clμ(El). By (2.1), we have μ(El⧹suppΛ(g))=0, which implies ∥Λ(g)−Λ(f)∥=∥Λ(g)−Λ(f)l∥+∥Λ(f)−Λ(f)l∥≥2−2∥Λ(f)l∥. Therefore, the first equality of (2.1) implies clμ(El)≤∥Λ(f)l∥. Furthermore, since ∑k=1nckμ(Ek)=1=∥Λ(f)∥=∑k=1n∥Λ(f)k∥, we conclude that ∫ElΛ(f)dμ=∥Λ(f)l∥=clμ(El)(l=1,…,n). Now, suppose that l is a fixed integer in {1,…,n} and Δ∈Ω satisfying Δ⊆El as well as μ(Δ)>0. If we set h≔Λ(1μ(Δ)χΔ)·χΔ, then the two equalities in Relation (2.1) imply that h=Λ(1μ(Δ)χΔ) as elements in L1(H) and that ∥Λ(f)−h∥=∥Λ(f)−Λ(χΔ/μ(Δ))∥=∥f−χΔ/μ(Δ)∥=2−2clμ(Δ). Therefore, Lemma 2.2(b) gives the required conclusion Λ(f)=f.□ For a W*-algebra M, we denote by (M) the set of projections. A map Ψ:(M)→(N) is called an orthoisomorphism if for any p,q∈(M), one has Ψ(p)·Ψ(q)=0 if and only if p·q=0. Proof of Proposition 2.1 Note that for normal states f and g of a W*-algebra with support projections sf and sg, respectively, one has ∥f−g∥=2ifandonlyifsf·sg=0. Thus, it follows from [16, Lemma 3.1(a)] that the metric preserving bijection Φ produces an orthoisomorphism Φˇ:(M)→(N) such that Φˇ(sf)=sΦ(f)(f∈(M*)+1). By the corollary in [6, p.18], we know that Φˇ extends to a Jordan *-isomorphism Θ:M→N, which is automatically weak- *-continuous (see, for example [18, Corollary 4.1.23]). Therefore, both M and N are abelian, and the map Θ is an *-isomorphism. Hence, M=L∞(X,Ω,μ) for a semi-finite measure space (X,Ω,μ) (see, for example [18, Proposition 1.18.1]), and its predual M* equals L1(X,Ω,μ). Consider Ψ≔Θ*−1∣L1(X,Ω,μ)+1. If we set Λ≔Ψ−1◦Φ, then Λ is a bijection from L1(X,Ω,μ)+1 onto itself satisfying the two relations in (2.1). Consequently, Lemma 2.3 tells us that Φ=Ψ as required.□ Now, we will give the proof of the parts of Theorem 1.1 concerning the invariants L1(G)+1 and M(G)+1. In fact, we have more precise statements for them as follows. In these statements, * is the convolution product. Theorem 2.4. Let Gand Hbe locally compact groups with Haar measures μGand μHthat define the norms on L1(G)and L1(H), respectively. (a) If Φ:L1(G)+1→L1(H)+1is a bijection satisfying Φ(f∗g)=Φ(f)∗Φ(g) and ∥Φ(f)−Φ(g)∥=∥f−g∥(f,g∈L1(G)+1), then there exist a homeomorphic group isomorphism ϕ:H→Gand a constant c>0such that Φ(f)(t)=cf(ϕ(t))for every f∈L1(G)+1and μH-almost every t∈H. (b) If Φ:M(G)+1→M(H)+1is a bijection satisfying Φ(α*β)=Φ(α)*Φ(β)and ∥Φ(α)−Φ(β)∥=∥α−β∥(α,β∈M(G)+1), then there exists a homeomorphic group isomorphism ϕ:H→Gsuch that Φ(α)(E)=α(ϕ(E)), for any α∈M(G)and compact subset E⊆G. Proof (a) Note that L1(G)+1 and L1(H)+1 are the normal state spaces of the abelian W*-algebras M=L∞(G) and N=L∞(H), respectively. From Proposition 2.1, we know that Φ can be extended to a surjective (complex) linear isometry from L1(G) onto L1(H). Now, the multiplicative assumption on Φ tells us that the extension is a Banach algebra isomorphism. By [24, Theorem 1], one obtains a homeomorphic group isomorphism ϕ:H→G, a continuous character θ:H→T and a constant c>0 satisfying Φ(f)(t)=cθ(t)f(ϕ(t)) for every f∈L1(G)+1 and μH-almost every t∈H. As Φ(χE)∈L1(G)+ for arbitrary measurable subset E⊆G with μG(E)=1, we know that θ(t)≥0 (or equivalently, θ(t)=1) for μH-almost every t∈H. Thus, the continuity of θ tells us that θ(t)=1 for all t∈H. (b) Note that M(G)+1 and M(H)+1 are the normal state spaces of the abelian W*-algebras C0(G)** and C0(H)**, respectively. Following the same line of argument as in part (a), but with [24, Theorem 1] being replaced by the paragraph following the Corollary in [10], we can find a homeomorphic group isomorphism ϕ:H→G and a continuous character θ:H→T with Φ(α)(E)=∫ϕ(E)θ(t)dα(t) for each α∈M(G) and each compact subset E⊆G. Since ∫ϕ(E)θ(t)dα(t)≥0 for every compact subset E⊆G and any α∈M(G)+, we know that θ(t)≥0 for μH-almost all t∈H. Consequently, θ(t)=1 for all t∈H.□ In order to present the other invariants in Theorem 1.1, we need to recall the notion of ‘left introverted subspace’ from [5] (see [13, 17] for more information). A closed subspace F of the C*-algebra Cb(G) of bounded continuous functions on a locally compact group G is said to be left introverted if for any s∈G, a∈F and f∈F*, one has λs(a)∈F; the function f⊙a:t↦f(λt(a)) belongs to F; here, λs(a)(t)≔a(s−1t) ( t∈G). In this case, F* is a Banach algebra under the product ⊙ defined by (f⊙g)(a)≔f(g⊙a) ( f,g∈F*;a∈F); see [5] for details. Suppose that A is a left introverted C*-subalgebra of Cb(G). It is not hard to check that (A*)+1 is closed under ⊙. Hence, (A*)+1 is a metric semigroup with the product ⊙. Examples of left introverted C*-subalgebras of Cb(G) are the space AP(G) of almost periodic continuous functions, the space WAP(G) of weakly almost periodic continuous functions, and the space LUC(G) of bounded left uniformly continuous functions. It follows from [20, Theorem 7] that LUC(G) is the largest left introverted closed subspace of Cb(G). Moreover, WAP(G) (respectively, AP(G)) is the largest left introverted closed subspaces of Cb(G) with the multiplication, ⊙, on the dual space being separately (respectively, jointly) weak- *-continuous on the unit sphere (see Theorems 5.6 and 5.8 of [12]). Theorem 2.5. Suppose that Aand Bare left introverted C*-subalgebras of Cb(G)and Cb(H)containing C0(G)and C0(H), respectively. If there is a bijection Φ:(A*)+1→(B*)+1satisfying Φ(f⊙g)=Φ(f)⊙Φ(g)and∥Φ(f)−Φ(g)∥=∥f−g∥(f,g∈(A*)+1),then Gand Hare isomorphic as topological groups. Proof Note that the double dual spaces A** and B** are both abelian W*-algebras. The argument is similar to that in the proof of Theorem 2.4(a), except that we need to use [13, Theorem 1] instead of [24, Theorem 1].□ It is easy to see that the left introverted C*-algebras WAP(G) and LUC(G) contain C0(G), and the remaining parts of Theorem 1.1 follow. Unlike WAP(G) and LUC(G), the intersection of the C*-subalgebra AP(G) with C0(G) is {0} unless G is compact. Thus, the argument for Theorem 2.5 does not work for AP(G). In fact, we have the following result. Let us recall some notation. As in [9], the almost periodic compactification (also known as the Bohr compactification), Gap, of a locally compact group G is the spectrum of the abelian C*-algebra AP(G), that is the weak- *-compact set of non-zero multiplicative linear functionals on AP(G). It is well known that Gap is a compact topological group under the weak- *-topology on AP(G)*. Corollary 2.6. Let Gand Hbe locally compact groups. Then (AP(G)*)+1≅(AP(H)*)+1as metric semigroups if and only if Gap≅Hapas topological groups. Proof It is well known that AP(G)≅C0(Gap) as ordered Banach algebras (see, for example [3, Section 4]). By Theorem 2.4(b) (notice that C0(Gap)*=M(Gap)), if (AP(G)*)+1≅(AP(H)*)+1, then Gap≅Hap. The converse is obvious.□ Note that the canonical group homomorphism sending G into Gap is not injective, unless AP(G) separates points of G; that is the case for example when G is either abelian or compact. In the most extreme situation, Gap is just a singleton set and such a group G is called minimally almost periodic in [21, 22]. For any minimally almost periodic group G, the metric semigroup (AP(G)*)+1 is the trivial one (that is contains only one element). 3. Further questions and investigations The Fourier algebra A(G) and the Fourier–Stieltjes algebra B(G) can be regarded as dual objects of L1(G) and M(G), respectively. In fact, in the framework of locally compact quantum groups, A(G) (respectively, B(G)) equals L1(Gˆ) (respectively, M(Gˆ)), where Gˆ is the ‘dual quantum group of G’ (which is not a locally compact group unless G is abelian). In [23], Walter showed that A(G) and B(G) are both complete invariants of G up to opposition. Some related results can be found in [1, 2, 7, 15, 17]. On the other hand, Walter’s result was extended to the quantum case by Daw and Le Pham (see [4]). It is natural to ask if A(G)+1 and B(G)+1 are also complete invariants of G up to opposition. Let us state this as a conjecture as follows. Conjecture 3.1. Let Gand Hbe locally compact groups, and Hopbe the opposite group of H. If there is a metric preserving semigroup isomorphism from A(G)+1 (respectively, B(G)+1) onto A(H)+1 (respectively, B(H)+1), then either G=Hor G=Hop. Recently, we have found a proof for the corresponding result of the above conjecture in the case of ‘type I’ locally compact quantum groups (see [14]). This can be used to obtain a positive answer for the above conjecture when G is either abelian or compact (or even when G is a compact quantum group). We are currently working on the general case. Funding The authors are partially supported by NSERC Grant ZC912, National Natural Science Foundation of China (11471168) and Taiwan MOST Grant (106-2115-M-110-006-MY2). References 1 W. Arendt and J. de Cannière , Order isomorphisms of Fourier algebras , J. Funct. Anal. 50 ( 1983 ), 1 – 7 . Google Scholar CrossRef Search ADS 2 W. 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Wong , A bounded semigroup invariant for some Banach algebras I: The type I case, preprint. 15 A. T.-M. Lau and N.-C. Wong , Orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups , J. Funct. Anal. 265 ( 2013 ), 562 – 593 . Google Scholar CrossRef Search ADS 16 C.-W. Leung , C.-K. Ng and N.-C. Wong , Transition probabilities of normal states determine the Jordan structure of a quantum system , J. Math. Phys. 57 ( 2016 ), 015212, 13 pages . doi:10.1063/1.4936404 . 17 P. L. Patterson , Characterizations of algebras arising from locally compact groups , Trans. Amer. Math. Soc. 329 ( 1992 ), 489 – 506 . Google Scholar CrossRef Search ADS 18 S. Sakai , C*-algebras and W*-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60 , Springer-Verlag , New York-Heidelberg , 1971 . 19 D. N. Tan , On extension of isometries on the unit spheres of Lp-spaces for 0 < p ≤ 1 , Nonlinear Anal. 74 ( 2011 ), 6981 – 6987 . 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Metric semigroups that determine locally compact groups

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Abstract

Abstract Let G be a locally compact group. Let A be any one of the (complex) Banach algebras: L1(G), M(G), WAP(G) and LUC(G), consisting of integrable functions, regular Borel complex measures, weakly almost periodic functions and bounded left uniformly continuous functions, respectively, on G. We show that the metric semigroup A+1:={f∈A:f≥0and∥f∥=1} (the convex structure is not considered) is a complete invariant for G. 1. Introduction In this paper, we find several new and simple complete invariants for locally compact groups. Let G and H be locally compact groups. Wendel showed in [24] (respectively, Johnson showed in [10]) that G and H are isomorphic if and only if there exists an isometric algebra isomorphism Φ:L1(G)→L1(H) (respectively, Φ:M(G)→M(H)). Optimistically, as Φ(sf)=sΦ(f), information in the one dimensional subspace {sf:s∈C} is somehow encoded in the element {f}. This leads to a quest of a ‘smaller invariant’. As a candidate, however, the unit sphere of L1(G) is not closed under the convolution product, and hence cannot be served as an invariant for G. On the other hand, Kawada showed in [11] that G and H are isomorphic whenever there is an algebra isomorphism Ψ:L1(G)→L1(H) satisfying: Ψ(f)≥0 if and only if f≥0. Observe that L1(G)+1, the positive part of the unit sphere of L1(G), is closed under the convolution product. This suggests us to consider L1(G)+1 as a candidate of a complete invariant of G. In this article, we will show that the metric and the semigroup structures of L1(G)+1, or those of M(G)+1 (note that the convexity is not needed), determine G. This result supplements the above-mentioned results of Wendel [24], Johnson [10] and Kawada [11]. Furthermore, Ghahramani et al. [8], as well as Lau and McKennon [13], showed that either one of the dual Banach algebras LUC(G)* and WAP(G)* determines G, too. We will also show that the positive parts of the unit spheres of LUC(G)* and WAP(G)* are complete invariants for G. For a subset S⊆E of an ordered Banach space E, we set S+1≔{f∈S:∥f∥=1;f≥0}. Our main results (namely, Theorems 2.4 and 2.5) can be subsumed and simplified in the following statement. Theorem 1.1. Two locally compact groups Gand Hare isomorphic as topological groups if and only if any one of the following holds: L1(G)+1≅L1(H)+1as metric semigroups; M(G)+1≅M(H)+1as metric semigroups; (WAP(G)*)+1≅(WAP(H)*)+1as metric semigroups; (LUC(G)*)+1≅(LUC(H)*)+1as metric semigroups. We will obtain the above assertions by verifying that those metric preserving semigroup isomorphisms actually extend to isometric algebra isomorphisms between the whole Banach algebras, and then the corresponding established results in [10, 11, 24] apply. This task is non-trivial. Although it has been shown in [19] that metric preserving bijection from the unit sphere of L1(G;R) (the space of real valued integrable functions) onto that of L1(H;R) extends to a real linear isometry from L1(G;R) onto L1(H;R), neither this statement nor the argument in [19] can be used in our cases. In fact, on top of elementary arguments, our proofs also depend on a theorem of Dye from [6] and its applications given in [16], which are results concerning W*-algebras. 2. The proof of the main theorem Theorem 1.1 is a consequence of the following result, which should be of independent interest (in particular, it tells us that the metric structure on the normal state space of a W*-algebra encodes its convex structure, when the algebra is abelian). Proposition 2.1. Let Mand Nbe W*-algebras, with one of them being abelian. If Φ:(M*)+1→(N*)+1is a bijection satisfying ∥Φ(f)−Φ(g)∥=∥f−g∥(f,g∈M), then there is a *-isomorphism Θ:M→Nsatisfying Θ*−1∣(M*)+1=Φ. Let us first do some preparation for the proof of Proposition 2.1. In the following, for any subset Δ of a set X, we denote by χΔ:X→{0,1} the characteristic function of Δ. Moreover, for any function g:X→C, we set supp g to be the support of g; namely, suppg≔{x∈X:g(x)≠0}. Lemma 2.2. Suppose that (X,Ω,μ)is a measure space and n∈N. (a) Let E∈Ωand c>0such that 0<cμ(E)≤1. Suppose that f∈L1(μ)+with suppf⊆Esuch that ∫Xfdμ=cμ(E). Then f=cχEif and only if for any Δ∈Ωwith Δ⊆Eand μ(Δ)>0, there exists gΔ∈L1(μ)+1satisfying suppgΔ⊆Δand ∥f−gΔ∥L1(μ)=1+cμ(E)−2cμ(Δ). (b) Let E1,…,En∈Ωand c1,…,cn>0satisfying ∑k=1nckμ(Ek)=1as well as Ei∩Ej=∅and μ(Ei)>0for any 1≤i≠j≤n. Consider f∈L1(μ)+1with ∫Elfdμ=clμ(El)(l=1,…,n).Then f=∑k=1nckχEkif and only if for any l∈{1,…,n}and Δ∈Ωwith Δ⊆Eland μ(Δ)>0, there exists hΔ,l∈L1(μ)+1satisfying supphΔ,l⊆Δand ∥f−hΔ,l∥L1(μ)=2−2clμ(Δ). Proof (a) ⇒). This implication is clear if we take gΔ≔1μ(Δ)χΔ. ⇐). For any r>0, we set Δr≔{x∈E:f(x)≤r}. Assume on the contrary that f≠cχE. Then one can find d∈(0,c) with μ(Δd)>0 (otherwise, f(x)≥c for μ-almost every x∈E, which, together with ∫Xfdμ=cμ(E), will imply f=cχE). Hence, we can find e∈(0,d] satisfying ∫Δdfdμ=eμ(Δd). Suppose gΔd∈L1(μ)+1 is as in the statement. Since suppgΔd⊆Δd, we know that ∥f−gΔd∥L1(μ)=∥f·χE⧹Δd∥L1(μ)+∥gΔd−f·χΔd∥L1(μ)≥∫E⧹Δdfdμ+(1−eμ(Δd))=1+cμ(E)−2eμ(Δd). This, together with the hypothesis, tells us that 2cμ(Δd)≤2eμ(Δd), and this contradicts with e<c. (b) ⇒). This implication is clear if we set hΔ,l≔1μ(Δ)χΔ. ⇐). Fix any l∈{1,…,n} and set fl≔χEl·f. Let Δ∈Ω with Δ⊆El and μ(Δ)>0. Consider hΔ,l∈L1(μ)+1 to be the element as in the statement. The equality ∥f−hΔ,l∥L1(μ)=∥f−fl∥L1(μ)+∥fl−hΔ,l∥L1(μ)=(1−clμ(El))+∥fl−hΔ,l∥L1(μ) implies ∥fl−hΔ,l∥L1(μ)=1+clμ(El)−2clμ(Δ). Thus, we conclude from part (a) that fl=clχEl. Since ∑k=1nckμ(Ek)=1 and ∫Xfdμ=1, we know that f=∑k=1nfk.□ Lemma 2.3. Let (X,Ω,μ)be a semi-finite measure space. If Λ:L1(μ)+1→L1(μ)+1is a bijection satisfying ∥Λ(f)−Λ(g)∥=∥f−g∥andμ(suppg⧹suppΛ(g))=0(f,g∈L1(μ)+1), (2.1)then Λis the identity map. Proof Since the set of positive simple functions with norm one is dense in L1(μ)+1, we only need to show that Λ(∑k=1nckχEk)=∑k=1nckχEk for any positive scalars c1,…,cn, and disjoint subsets E1,…,En∈Ω with μ(Ei)>0 ( i=1,…,n) satisfying ∑k=1nckμ(Ek)=1. Let us set f≔∑k=1nckχEk and fix an arbitrary integer l∈{1,…,n}. Let us also denote Λ(f)l≔Λ(f)·χElandg≔χEl/μ(El). Then ∥f−g∥L1(μ)=2−2clμ(El). By (2.1), we have μ(El⧹suppΛ(g))=0, which implies ∥Λ(g)−Λ(f)∥=∥Λ(g)−Λ(f)l∥+∥Λ(f)−Λ(f)l∥≥2−2∥Λ(f)l∥. Therefore, the first equality of (2.1) implies clμ(El)≤∥Λ(f)l∥. Furthermore, since ∑k=1nckμ(Ek)=1=∥Λ(f)∥=∑k=1n∥Λ(f)k∥, we conclude that ∫ElΛ(f)dμ=∥Λ(f)l∥=clμ(El)(l=1,…,n). Now, suppose that l is a fixed integer in {1,…,n} and Δ∈Ω satisfying Δ⊆El as well as μ(Δ)>0. If we set h≔Λ(1μ(Δ)χΔ)·χΔ, then the two equalities in Relation (2.1) imply that h=Λ(1μ(Δ)χΔ) as elements in L1(H) and that ∥Λ(f)−h∥=∥Λ(f)−Λ(χΔ/μ(Δ))∥=∥f−χΔ/μ(Δ)∥=2−2clμ(Δ). Therefore, Lemma 2.2(b) gives the required conclusion Λ(f)=f.□ For a W*-algebra M, we denote by (M) the set of projections. A map Ψ:(M)→(N) is called an orthoisomorphism if for any p,q∈(M), one has Ψ(p)·Ψ(q)=0 if and only if p·q=0. Proof of Proposition 2.1 Note that for normal states f and g of a W*-algebra with support projections sf and sg, respectively, one has ∥f−g∥=2ifandonlyifsf·sg=0. Thus, it follows from [16, Lemma 3.1(a)] that the metric preserving bijection Φ produces an orthoisomorphism Φˇ:(M)→(N) such that Φˇ(sf)=sΦ(f)(f∈(M*)+1). By the corollary in [6, p.18], we know that Φˇ extends to a Jordan *-isomorphism Θ:M→N, which is automatically weak- *-continuous (see, for example [18, Corollary 4.1.23]). Therefore, both M and N are abelian, and the map Θ is an *-isomorphism. Hence, M=L∞(X,Ω,μ) for a semi-finite measure space (X,Ω,μ) (see, for example [18, Proposition 1.18.1]), and its predual M* equals L1(X,Ω,μ). Consider Ψ≔Θ*−1∣L1(X,Ω,μ)+1. If we set Λ≔Ψ−1◦Φ, then Λ is a bijection from L1(X,Ω,μ)+1 onto itself satisfying the two relations in (2.1). Consequently, Lemma 2.3 tells us that Φ=Ψ as required.□ Now, we will give the proof of the parts of Theorem 1.1 concerning the invariants L1(G)+1 and M(G)+1. In fact, we have more precise statements for them as follows. In these statements, * is the convolution product. Theorem 2.4. Let Gand Hbe locally compact groups with Haar measures μGand μHthat define the norms on L1(G)and L1(H), respectively. (a) If Φ:L1(G)+1→L1(H)+1is a bijection satisfying Φ(f∗g)=Φ(f)∗Φ(g) and ∥Φ(f)−Φ(g)∥=∥f−g∥(f,g∈L1(G)+1), then there exist a homeomorphic group isomorphism ϕ:H→Gand a constant c>0such that Φ(f)(t)=cf(ϕ(t))for every f∈L1(G)+1and μH-almost every t∈H. (b) If Φ:M(G)+1→M(H)+1is a bijection satisfying Φ(α*β)=Φ(α)*Φ(β)and ∥Φ(α)−Φ(β)∥=∥α−β∥(α,β∈M(G)+1), then there exists a homeomorphic group isomorphism ϕ:H→Gsuch that Φ(α)(E)=α(ϕ(E)), for any α∈M(G)and compact subset E⊆G. Proof (a) Note that L1(G)+1 and L1(H)+1 are the normal state spaces of the abelian W*-algebras M=L∞(G) and N=L∞(H), respectively. From Proposition 2.1, we know that Φ can be extended to a surjective (complex) linear isometry from L1(G) onto L1(H). Now, the multiplicative assumption on Φ tells us that the extension is a Banach algebra isomorphism. By [24, Theorem 1], one obtains a homeomorphic group isomorphism ϕ:H→G, a continuous character θ:H→T and a constant c>0 satisfying Φ(f)(t)=cθ(t)f(ϕ(t)) for every f∈L1(G)+1 and μH-almost every t∈H. As Φ(χE)∈L1(G)+ for arbitrary measurable subset E⊆G with μG(E)=1, we know that θ(t)≥0 (or equivalently, θ(t)=1) for μH-almost every t∈H. Thus, the continuity of θ tells us that θ(t)=1 for all t∈H. (b) Note that M(G)+1 and M(H)+1 are the normal state spaces of the abelian W*-algebras C0(G)** and C0(H)**, respectively. Following the same line of argument as in part (a), but with [24, Theorem 1] being replaced by the paragraph following the Corollary in [10], we can find a homeomorphic group isomorphism ϕ:H→G and a continuous character θ:H→T with Φ(α)(E)=∫ϕ(E)θ(t)dα(t) for each α∈M(G) and each compact subset E⊆G. Since ∫ϕ(E)θ(t)dα(t)≥0 for every compact subset E⊆G and any α∈M(G)+, we know that θ(t)≥0 for μH-almost all t∈H. Consequently, θ(t)=1 for all t∈H.□ In order to present the other invariants in Theorem 1.1, we need to recall the notion of ‘left introverted subspace’ from [5] (see [13, 17] for more information). A closed subspace F of the C*-algebra Cb(G) of bounded continuous functions on a locally compact group G is said to be left introverted if for any s∈G, a∈F and f∈F*, one has λs(a)∈F; the function f⊙a:t↦f(λt(a)) belongs to F; here, λs(a)(t)≔a(s−1t) ( t∈G). In this case, F* is a Banach algebra under the product ⊙ defined by (f⊙g)(a)≔f(g⊙a) ( f,g∈F*;a∈F); see [5] for details. Suppose that A is a left introverted C*-subalgebra of Cb(G). It is not hard to check that (A*)+1 is closed under ⊙. Hence, (A*)+1 is a metric semigroup with the product ⊙. Examples of left introverted C*-subalgebras of Cb(G) are the space AP(G) of almost periodic continuous functions, the space WAP(G) of weakly almost periodic continuous functions, and the space LUC(G) of bounded left uniformly continuous functions. It follows from [20, Theorem 7] that LUC(G) is the largest left introverted closed subspace of Cb(G). Moreover, WAP(G) (respectively, AP(G)) is the largest left introverted closed subspaces of Cb(G) with the multiplication, ⊙, on the dual space being separately (respectively, jointly) weak- *-continuous on the unit sphere (see Theorems 5.6 and 5.8 of [12]). Theorem 2.5. Suppose that Aand Bare left introverted C*-subalgebras of Cb(G)and Cb(H)containing C0(G)and C0(H), respectively. If there is a bijection Φ:(A*)+1→(B*)+1satisfying Φ(f⊙g)=Φ(f)⊙Φ(g)and∥Φ(f)−Φ(g)∥=∥f−g∥(f,g∈(A*)+1),then Gand Hare isomorphic as topological groups. Proof Note that the double dual spaces A** and B** are both abelian W*-algebras. The argument is similar to that in the proof of Theorem 2.4(a), except that we need to use [13, Theorem 1] instead of [24, Theorem 1].□ It is easy to see that the left introverted C*-algebras WAP(G) and LUC(G) contain C0(G), and the remaining parts of Theorem 1.1 follow. Unlike WAP(G) and LUC(G), the intersection of the C*-subalgebra AP(G) with C0(G) is {0} unless G is compact. Thus, the argument for Theorem 2.5 does not work for AP(G). In fact, we have the following result. Let us recall some notation. As in [9], the almost periodic compactification (also known as the Bohr compactification), Gap, of a locally compact group G is the spectrum of the abelian C*-algebra AP(G), that is the weak- *-compact set of non-zero multiplicative linear functionals on AP(G). It is well known that Gap is a compact topological group under the weak- *-topology on AP(G)*. Corollary 2.6. Let Gand Hbe locally compact groups. Then (AP(G)*)+1≅(AP(H)*)+1as metric semigroups if and only if Gap≅Hapas topological groups. Proof It is well known that AP(G)≅C0(Gap) as ordered Banach algebras (see, for example [3, Section 4]). By Theorem 2.4(b) (notice that C0(Gap)*=M(Gap)), if (AP(G)*)+1≅(AP(H)*)+1, then Gap≅Hap. The converse is obvious.□ Note that the canonical group homomorphism sending G into Gap is not injective, unless AP(G) separates points of G; that is the case for example when G is either abelian or compact. In the most extreme situation, Gap is just a singleton set and such a group G is called minimally almost periodic in [21, 22]. For any minimally almost periodic group G, the metric semigroup (AP(G)*)+1 is the trivial one (that is contains only one element). 3. Further questions and investigations The Fourier algebra A(G) and the Fourier–Stieltjes algebra B(G) can be regarded as dual objects of L1(G) and M(G), respectively. In fact, in the framework of locally compact quantum groups, A(G) (respectively, B(G)) equals L1(Gˆ) (respectively, M(Gˆ)), where Gˆ is the ‘dual quantum group of G’ (which is not a locally compact group unless G is abelian). In [23], Walter showed that A(G) and B(G) are both complete invariants of G up to opposition. Some related results can be found in [1, 2, 7, 15, 17]. On the other hand, Walter’s result was extended to the quantum case by Daw and Le Pham (see [4]). It is natural to ask if A(G)+1 and B(G)+1 are also complete invariants of G up to opposition. Let us state this as a conjecture as follows. Conjecture 3.1. Let Gand Hbe locally compact groups, and Hopbe the opposite group of H. If there is a metric preserving semigroup isomorphism from A(G)+1 (respectively, B(G)+1) onto A(H)+1 (respectively, B(H)+1), then either G=Hor G=Hop. Recently, we have found a proof for the corresponding result of the above conjecture in the case of ‘type I’ locally compact quantum groups (see [14]). This can be used to obtain a positive answer for the above conjecture when G is either abelian or compact (or even when G is a compact quantum group). We are currently working on the general case. Funding The authors are partially supported by NSERC Grant ZC912, National Natural Science Foundation of China (11471168) and Taiwan MOST Grant (106-2115-M-110-006-MY2). References 1 W. Arendt and J. de Cannière , Order isomorphisms of Fourier algebras , J. Funct. Anal. 50 ( 1983 ), 1 – 7 . Google Scholar CrossRef Search ADS 2 W. 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Wong , A bounded semigroup invariant for some Banach algebras I: The type I case, preprint. 15 A. T.-M. Lau and N.-C. Wong , Orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups , J. Funct. Anal. 265 ( 2013 ), 562 – 593 . Google Scholar CrossRef Search ADS 16 C.-W. Leung , C.-K. Ng and N.-C. Wong , Transition probabilities of normal states determine the Jordan structure of a quantum system , J. Math. Phys. 57 ( 2016 ), 015212, 13 pages . doi:10.1063/1.4936404 . 17 P. L. Patterson , Characterizations of algebras arising from locally compact groups , Trans. Amer. Math. Soc. 329 ( 1992 ), 489 – 506 . Google Scholar CrossRef Search ADS 18 S. Sakai , C*-algebras and W*-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60 , Springer-Verlag , New York-Heidelberg , 1971 . 19 D. N. Tan , On extension of isometries on the unit spheres of Lp-spaces for 0 < p ≤ 1 , Nonlinear Anal. 74 ( 2011 ), 6981 – 6987 . 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Published: Nov 28, 2017

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