Group algebras with a unique C∗-norm

Boidol, Joachim

doi:10.1016/0022-1236(84)90088-0

Cited by 14 publications

(14 citation statements)

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“…One of the permanence properties of C Ã -unique groups mentioned in [7] is the following (see the proof of [7, Theorem 1]): if G is a C Ã -unique group and K is a compact normal subgroup of G; then G=K is also C Ã -unique. This can be shown by using condition (U4) in Section 1 and the following fact: for each compact normal subgroup K; d G=K G=K can be regarded as an open subset ofĜ (see [19, 5.2]).…”

Section: Some Permanence Properties Of C ã -Unique Groupsmentioning

confidence: 99%

“…In fact, if jj Á jj 0 is a natural norm on L 1 ðNÞ in the sense of [7,Section 5] and m is the injective Ã-representation defining jj Á jj 0 ; then by Lemma 2.5, we have jjind G N ðmÞðhÞjj ¼ jjmðhÞjj (as a x is a Ã-automorphism on L 1 ðNÞ) and the same argument as for Theorem 2.7(b) shows that jj Á jj 0 coincides with the norm induced from C Ã ðNÞ: Note also that in the case of connected locally compact group, weakly C Ã -uniqueness is the same as amenability [7,Theorem 5]. Whether this holds for more general groups is unknown.…”

Section: Article In Pressmentioning

confidence: 99%

“…For each element f AL 1 ðGÞ;f is the Fourier transform of f : C Ã -unique groups were first introduced and studied by Boidol [7]. A locally compact group G is said to be C Ã -unique if (U1) L 1 ðGÞ has a unique C Ã -norm.…”

Section: Introductionmentioning

confidence: 99%

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Some permanence properties of C∗-unique groups

Leung

2004

Journal of Functional Analysis

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We will study some permanence properties of C Ã -unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C Ã -unique group. Moreover, any amenable discrete group is a retract of a discrete C Ã -unique group. r 2003 Elsevier Inc. All rights reserved. MSC: primary 43A20; 22D25; secondary 22D30; 22D20

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Section: Some Permanence Properties Of C ã -Unique Groupsmentioning

confidence: 99%

Section: Article In Pressmentioning

confidence: 99%

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Some permanence properties of C∗-unique groups

Leung

2004

Journal of Functional Analysis

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“…Moreover, it is easy to see that one has the corresponding results for Theorem 9 and Corollary 11 for twisted covariant systems (in this case, almost-freeness means that t∈G\N Prim(A) t is nowhere dense in Prim(A)). Using this, one can show that if G is a separable group with an abelian closed normal subgroup N such that t∈G\NN t is nowhere dense in the dual groupN , then G is a C * -unique group (in the sense of [2]; see [10] for the details).…”

Section: Corollary 11mentioning

confidence: 99%

“…(b) Note that if the action of G on A is quasi-regular (see [4, p.223]) and every quasi-orbit is regular (see [4, p.223]), then by using [4, propostion 20] instead of [7, 3.2], one can remove the separabilities for both G and A in Lemma 3 as well as in Theorem 9 and obtain certain untwisted versions of Theorem 9 and Corollary 11 (in this case, we only have (1) being equivalent to (2) and (3) being equivalent to (4)). …”

Section: Corollary 11mentioning

confidence: 99%

Invariant ideals of twisted crossed products

Leung

2003

Mathematische Zeitschrift

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We will prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on C * -algebras. Mathematics Subject Classification (2000): 46L05, 46L55The motivation of this study is our recent research on C * -unique groups in [10]. In fact, one interesting question is when the semi-direct product of a C * -unique group with another group is again C * -unique. This turns out to be related to the following question: Given a C * -dynamical system (A, G, α), under what condition will it be true that any non-zero ideal of A × α G contains a non-zeroα-invariant ideal (α being the dual coaction)? The aim of this paper is to study this question.In fact, in the case of discrete amenable groups acting on compact spaces, Kawamura and Tomiyama gave (in [9]) a complete solution of the above question. In [1], Archbold and Spielberg generalised the main result in [9] to the case of discrete C * -dynamical system. In this article, we are going to present a weaker result but in the case of general locally compact groups. As a corollary, we obtain some equivalent conditions for the primeness of crossed products (in terms of the actions). Moreover, we will also give a brief discussion on the simplicity of crossed products (which is related but does not need the main theorem).

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Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

Austad

2021

J Fourier Anal Appl

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We show spectral invariance for faithful $$*$$ ∗ -representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group $$G_c$$ G c is $$C^*$$ C ∗ -unique and symmetric, then the twisted convolution algebra $$L^1 (G,c)$$ L 1 ( G , c ) is spectrally invariant in $${\mathbb {B}}({\mathcal {H}})$$ B ( H ) for any faithful $$*$$ ∗ -representation of $$L^1 (G,c)$$ L 1 ( G , c ) as bounded operators on a Hilbert space $${\mathcal {H}}$$ H . As an application of this result we give a proof of the statement that if $$\Delta $$ Δ is a closed cocompact subgroup of the phase space of a locally compact abelian group $$G'$$ G ′ , and if g is some function in the Feichtinger algebra $$S_0 (G')$$ S 0 ( G ′ ) that generates a Gabor frame for $$L^2 (G')$$ L 2 ( G ′ ) over $$\Delta $$ Δ , then both the canonical dual atom and the canonical tight atom associated to g are also in $$S_0 (G')$$ S 0 ( G ′ ) . We do this without the use of periodization techniques from Gabor analysis.

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Group algebras with a unique C∗-norm

Cited by 14 publications

References 8 publications

Some permanence properties of C∗-unique groups

Some permanence properties of C∗-unique groups

Invariant ideals of twisted crossed products

Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

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