Finite-Dimensional Representations of Free Product C*-Algebras

Exel, Ruy; Loring, Terry A.

doi:10.1142/s0129167x92000217

Cited by 61 publications

(75 citation statements)

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“…The first result linking full free products and residual finite dimensionality was M.-D. Choi's proof [6] that the full group C * -algebras of nonabelian free groups are r.f.d. In [7], Exel and Loring proved that the full free product of any two r.f.d. C * -algebras A and B with amalgamation over either the zero C * -algebra or over the scalar multiples of the identity (if A and B are unital) is r.f.d.…”

Section: Corollary 34 Supposementioning

confidence: 99%

On embeddings of full amalgamated free product C*–algebras

Armstrong

Dykema

Exel

et al. 2004

Proc. Amer. Math. Soc.

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Abstract. We examine the question of when the * -homomorphism λ : A * D B → A * D B of full amalgamated free product C * -algebras, arising from compatible inclusions of C * -algebras A ⊆ A, B ⊆ B and D ⊆ D, is an embedding. Results giving sufficient conditions for λ to be injective, as well as classes of examples where λ fails to be injective, are obtained. As an application, we give necessary and sufficient conditions for the full amalgamated free product of finite-dimensional C * -algebras to be residually finite dimensional.

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Section: Corollary 34 Supposementioning

confidence: 99%

On embeddings of full amalgamated free product C*–algebras

Armstrong

Dykema

Exel

et al. 2004

Proc. Amer. Math. Soc.

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“…The category of C * -algebras has coproducts, and then, according to [11,Theorem 3.2], Proposition 2.7 holds verbatim for C * -algebras A and B. We will use that result in the proof of Proposition 2.7, but indirectly.…”

Section: 2mentioning

confidence: 99%

“…We would be done if we knew that A i * B j are RFD, but now we can make use of the Exel-Loring result cited above: Being finite-dimensional C * -algebras, A i and B j are RFD in the C * sense, and hence [11,Theorem 3.2] applies to them. It implies that the C * -completion is RFD in the C * sense, and it is easy to see in this case that the * -algebra A i * B j embeds in its C * -completion.…”

Section: 2mentioning

confidence: 99%

“…In [12] for instance, the authors prove among other things that the full C * -algebra of a monoid that is a coproduct (in the category of monoids) of free groups and free or finite monoids is RFD; and in [11] Exel and Loring show that the coproduct of two RFD C * -algebras is again RFD, generalizing all of the previously-cited results (the full C * -algebra C * (F 2 ) of F 2 for instance is the coproduct of two copies of the RFD full C * -algebra C * (Z)).…”

Section: Introductionmentioning

confidence: 99%

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Residually finite quantum group algebras

Chirvăsitu¹

2015

Journal of Functional Analysis

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Abstract. We show that provided n = 3, the involutive Hopf * -algebra Au(n) coacting universally on an n-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in some finite-dimensional * -representation. This implies that the discrete quantum group with group algebra Au(n) is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. So ltan in [19].We also prove analogous results for the involutive Hopf * -algebra Bu(n) coacting universally on an n-dimensional Hilbert space equipped with a non-degenerate bilinear form. IntroductionThis paper is concerned with the property of residual finite-dimensionality (or RFD for short) for operator algebras associated to discrete quantum groups. For a C * -algebra A being RFD means having a separating family representations on finite-dimensional Hilbert spaces; in other words, for any 0 = a ∈ A there is a C * -algebra homomorphism π : A → M n (C) that does not annihilate a.The RFD property is in a sense the exact opposite of simplicity: A C * -algebra is simple if it has no non-obvious quotients, whereas it is RFD if it has plenty of small quotients. The free group F 2 on two generators is the perfect example for both extremes: It is a standard result that its so-called reduced C * -algebra (i.e. the closure in B(ℓ 2 (F 2 )) of the left regular representation) is simple, while the C * -algebra universally generated by two unitaries (the full C * -algebra of F 2 ) is RFD [9].In fact freeness of some sort has been central in investigating the RFD property. In [12] for instance, the authors prove among other things that the full C * -algebra of a monoid that is a coproduct (in the category of monoids) of free groups and free or finite monoids is RFD; and in [11] Exel and Loring show that the coproduct of two RFD C * -algebras is again RFD, generalizing all of the previously-cited results (the full C * -algebra C * (F 2 ) of F 2 for instance is the coproduct of two copies of the RFD full C * -algebra C * (Z)).All of the above references deal extensively with group C * -algebras of discrete groups. The same kinds of issues are raised in [19] in the context of discrete quantum groups, where the main objects under consideration are the so-called CQG algebras of [10].We recall below ( §2.1) that these are algebras (in fact Hopf algebras) which should be thought of as comprising the representative functions on a compact "quantum group". By a kind of noncommutative Pontryagin duality, such an algebra is also trying to be the group algebra of a discrete quantum group. Just as for a classical discrete group, a CQG algebra has two extremal completions to a C * -algebra: a largest one called 'full' and a smallest one called 'reduced'. The paper [19] defines and constructs Bohr compactifications for discrete quantum groups. The procedure is parallel to the classical one of compactifying an ordinary discrete group, and many 2010 Mathematics Subject Classifica...

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“…Thierry Giordano brought the useful paper [8] to our attention. Terry Loring informed us of [11]. The first author benefited from conversations with Vladimir Platonov on free products.…”

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confidence: 99%

Diagonal type conditions on group C*-algebras

Spronk

2000

Proc. Amer. Math. Soc.

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Let G G be a locally compact group with C ∗ ( G ) \mathbf {C}^* (G) and C r ∗ ( G ) \mathbf {C}^*_r (G) its enveloping and reduced C ∗ ^* -algebras respectively. We show that if C ∗ ( G ) \mathbf {C}^*(G) is residually finite dimensional, then G G is maximally almost periodic, and C r ∗ ( G ) \mathbf {C}^*_r (G) is residually finite dimensional if and only if G G is both amenable and maximally almost periodic. Letting λ G \lambda _G be the left regular representation of G G , we show that a certain quasidiagonality condition on { λ G ( s ) : s ∈ G } \{\lambda _G(s):s\in G\} implies that G G is amenable.

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Finite-Dimensional Representations of Free Product C*-Algebras

Cited by 61 publications

References 4 publications

On embeddings of full amalgamated free product C*–algebras

On embeddings of full amalgamated free product C*–algebras

Residually finite quantum group algebras

Diagonal type conditions on group C*-algebras

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