Maximality of dual coactions on sectional $C^{*}$-algebras of Fell bundles and applications

Buss, Alcides; Echterhoff, Siegfried

doi:10.4064/sm8361-1-2016

Cited by 4 publications

(8 citation statements)

References 65 publications

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“…For 1 ≤ p ≤ ∞, the L p -C*-algebra C * L p (G) is the completion of L 1 (G) with respect to the C*-norm • C * L p (G) defined by f C * L p (G) = sup{ π(f ) : π is an L p -representation of G}. The L p -C*-algebras for locally compact groups were first introduced by Brown and Guentner in the case of discrete groups (see [6]) and have since been studied by many different authors (see [5,7,8,9,15,20,21,22,23,30,32,35,36,37]). We list a few properties these C*-algebras possess.…”

Section: Introductionmentioning

confidence: 99%

Exotic C*-algebras of geometric groups

Samei¹,

Wiersma²

2018

Preprint

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We consider a new class of potentially exotic group C*-algebras C * (PF * p (G)) for a locally compact group G, and its connection with the class of potentially exotic group C*algebras C * L p (G) introduced by Brown and Guentner. Surprisingly, these two classes of C*algebras are intimately related. By exploiting this connection, we show C * L p (G) = C * (PF * p (G)) for p ∈ (2, ∞), and the C*-algebras C * L p (G) are pairwise distinct for p ∈ (2, ∞) when G belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of C * L p (G) and C * (PF * p (G)). This greatly generalizes earlier results of Okayasu (see [30]) and the second author (see [37]) on the pairwise distinctness of C * L p (G) for 2 < p < ∞ when G is either a noncommutative free group or the group SL(2, R), respectively.As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group G. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of G in the left regular representation of G (see [14]). Also we give a near solution to a 1978 conjecture of Cowling stated in [10]. This conjecture of Cowling states if G is a Kunze-Stein group and π is a unitary representation of G with cyclic vector ξ such that the map G ∋ s → π(s)ξ, ξ belongs to L p (G) for some 2 < p < ∞, then Aπ ⊆ L p (G). We show Bπ ⊆ L p+ǫ (G) for every ǫ > 0 (recall Aπ ⊆ Bπ).

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Section: Introductionmentioning

confidence: 99%

Exotic C*-algebras of geometric groups

Samei¹,

Wiersma²

2018

Preprint

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“…Remark 4.4. The above result extends to the exotic cross-sectional C * -algebras C * µ (B) associated to Morita compatible cross-product functors ⋊ µ as defined in [7]. This is because, by definition, C * µ (B) is the quotient of C * (B) that turns the isomorphism (3.4) into an isomorphism…”

Section: Morita Equivalence Of Actions and Fell Bundlesmentioning

confidence: 57%

“…As a consequence of this, the notion of weak (hence strong) equivalence preserves full and reduced cross-sectional C * -algebras, that is, weakly equivalent Fell bundles have (strongly Morita) equivalent full and reduced cross-sectional C * -algebras. Using the same idea, we also derive a version of this result for certain exotic completions C * µ (A) introduced in [7] (some of the latter results were also obtained in [4], though with different methods). Moreover, we show that two Fell bundles A and B are weakly equivalent if and only if the corresponding actions on their C * -algebras of kernels (A) and (B) are equivariantly Morita equivalent, if and only if their dual coactions on C * (A) and C * (B) are equivariantly Morita equivalent.…”

Section: Introductionmentioning

confidence: 88%

“…Moreover, this isomorphism carries the canonical action of G on (B) to the dual action of G on C * (B) ⋊ δB G. Thus (B) ∼ = C * (B) ⋊ δB G as G-C * -algebras. The dual coaction on C * (B) is maximal and its normalisation is the dual coaction δ r B on C * r (B) (see [7]). This means that there exists a natural isomorphism…”

Section: This Implies Inequality (33)mentioning

confidence: 99%

“…And this isomorphism preserves the dual coactions whenever the exotic crossedproduct admits such a coaction; this means that ⋊ µ is a duality cross-product functor in the language of [8] (see also [4]). This is a big class of functors and, in particular, includes all correspondence functors (see [7,Corollary 4.6]). We will have more to say about amenability in Section 6.…”

Section: Morita Equivalence Of Actions and Fell Bundlesmentioning

confidence: 99%

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Morita Enveloping Fell Bundles

Abadie

Buss

Ferraro

2018

Bull Braz Math Soc, New Series

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We introduce notions of weak and strong equivalence for nonsaturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional C * -algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.

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Exotic Crossed Products

Buss

Echterhoff

Willett

2016

Operator Algebras and Applications

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An exotic crossed product is a way of associating a C * -algebra to each C * -dynamical system that generalizes the well-known universal and reduced crossed products. Exotic crossed products provide natural generalizations of, and tools to study, exotic group C * -algebras as recently considered by Brown-Guentner and others. They also form an essential part of a recent program to reformulate the Baum-Connes conjecture with coefficients so as to mollify the counterexamples caused by failures of exactness.In this paper, we survey some constructions of exotic group algebras and exotic crossed products. Summarising our earlier work, we single out a large class of crossed products -the correspondence functors -that have many properties known for the maximal and reduced crossed products: for example, they extend to categories of equivariant correspondences, and have a compatible descent morphism in KK-theory. Combined with known results on K-amenability and the Baum-Connes conjecture, this allows us to compute the K-theory of many exotic group algebras. It also gives new information about the reformulation of the Baum-Connes Conjecture mentioned above. Finally, we present some new results relating exotic crossed products for a group and its closed subgroups, and discuss connections with the reformulated Baum-Connes conjecture.2010 Mathematics Subject Classification. 46L55, 46L08, 46L80.

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Maximality of dual coactions on sectional $C^{*}$-algebras of Fell bundles and applications

Cited by 4 publications

References 65 publications

Exotic C*-algebras of geometric groups

Exotic C*-algebras of geometric groups

Morita Enveloping Fell Bundles

Exotic Crossed Products

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