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Exotic Crossed Products
Operator Algebras · Submitted 9 Oct 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.07 | 0.04 | asserted 1.00 |
| s1 | Exotic crossed products provide natural generalizations of, and tools to study, exotic group C*-algebras as recently considered by Brown-Guentner and others. | background 0.94 | 0.12 | 0.05 | asserted 1.00 |
| s2 | 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. | background 0.99 | 0.08 | 0.05 | asserted 1.00 |
| s3 | In this paper, we survey some constructions of exotic group algebras and exotic crossed products. | contribution 0.94 | 0.62 | 0.05 | asserted 0.93 |
| s4 | 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. | contribution 0.97 | 0.86 | 0.37 | asserted 1.00 |
| s5 | 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. | result 0.95 | 0.85 | 0.21 | asserted 0.99 |
| s6 | It also gives new information about the reformulation of the Baum-Connes Conjecture mentioned above. | result 0.70 | 0.79 | 0.13 | asserted 0.98 |
| s7 | 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. | contribution 0.97 | 0.91 | 0.17 | asserted 0.96 |
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Original abstract
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.
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How the literature uses it
4 arXiv papers cite this in 12 sentences
Papers citing it · 8 sentences without a confident label
Compares with it 1 paper
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Thus [equation] This argument should be compared to [ref] and [ref] .
Ordered invariant ideals of Fourier-Stieltjes algebras · Investigation of the question · 1.00
Cites it as background 3 papers
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We also recall another type of coaction functor: decreasing, which include those coaction functors arising from large ideals of the Fourier-Stieltjes algebra $B(G)$; the associated crossed-product functors for actions have been referred to as “KLQ functors” [ref] or “KLQ crossed products” [ref] .
Coaction functors, II · Introduction · 0.98
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Some effort has been expended to “fix” the Baum-Connes conjecture (see, e.g., [ref] ).
Exact sequences in the Enchilada category · Introduction · 0.87
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Recently there has been a flurry of activity regarding constructions using these group $C^*$-algebras (a brief sampling: [ref] ).
Ordered invariant ideals of Fourier-Stieltjes algebras · Introduction · 0.99
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