Coaction Functors and Exact Large Ideals of Fourier-Stieltjes Algebras

Transcription

1 Coaction Functors and Exact Large Ideals of Fourier-Stieltjes Algebras S. Kaliszewski*, Arizona State University Magnus Landstad, NTNU John Quigg, ASU AMS Fall Southeastern Sectional Meeting Special Session on the Analysis, Geometry, and Topology of Groupoids University of Memphis, TN October 17, / 58

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3 Thank You! 3 / 58

4 References 1. Brown-Guentner, New C -completions of discrete groups and related spaces, arxiv: K-Landstad-Quigg, Exotic group C -algebras in noncommutative duality, arxiv: , Exotic coactions, arxiv: Baum-Guentner-Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, arxiv: Buss-Echterhoff-Willett, Exotic crossed products and the Baum-Connes conjecture, arxiv: K-Landstad-Quigg, Coaction functors, arxiv: , Exact large ideals of B(G) are downward directed, arxiv: Buss-Echterhoff-Willett, Exotic crossed products, arxiv:1510: / 58

5 Any Questions? Does every crossed product functor come from a coaction functor? Does every coaction functor come from some large ideal? (No.) Does every exact one come from some exact large ideal? Which large ideals are exact? If G is exact and 2 p <, is E p = B(G) L p (G) exact? 5 / 58

6 Conclusion For any large ideal E of B(G), the assignment is functorial on Ĝ-C. τ E : (A, δ) (A E, δ E ) There are surjective natural transformations (A m, δ m ) (A E, δ E ) (A n, δ n ). (A E, δ E ) is Morita equivalent to (B E, ɛ E ) whenever (A, δ) is Morita equivalent to (B, ɛ). τ E preserves short exact sequences (E is exact) if and only if {a A E a I } I + {a A E a = 0} for all (A, δ) and strongly invariant I A. If E and F are both exact, then E F is also exact. 6 / 58

7 Large Ideals of B(G) An ideal E of B(G) = C (G) C b (G) is large if E is weak- closed, contains B r (G) = C r (G), and is invariant under the natural action(s) of G on B(G): (x f y)(z) = f (xzy) large ideal E of B(G) small ideal E of C (G) exotic group C -algebra CE (G) = C (G)/ E with coaction δg E of G 7 / 58

8 Large Ideals of B(G) Given a coaction (A, δ) of G: large ideal E small δ-invariant ideal A E = {a A E a = 0} an exotic coaction δ E of G on A E = A/A E τ Br (G) is normalization (A, δ) (A n, δ n ). τ B(G) is the identity functor (A, δ) (A, δ). Maximalization (A, δ) (A m, δ m ) is not τ E for any E. 8 / 58

9 The Baum-Connes Conjecture(s) For any second-countable locally compact group G, and any G-C -algebra A, the (reduced) assembly map is an isomorphism. µ red : K top (G; A) K (A red G) Counterexamples... are closely connected to failures of exactness Maybe the maximal assembly map is an isomorphism. µ max : K top (G; A) K (A max G) There are well-known Property (T) obstructions... 9 / 58

10 Crossed Product Functors Study crossed products that combine the good properties of the maximal and reduced crossed products. A crossed product is a functor A A τ G from G-C to C, together with natural transformations A max G A τ G A red G restricting to the identity map on the dense subalgebra(s) A alg G. Each has a τ-assembly map µ τ : K top (G; A) K (A max G) K (A τ G). 10 / 58

11 Crossed Product Functors τ is exact if 0 I τ G A τ G B τ G 0 is short exact in C whenever 0 I A B 0 is short exact in G-C. τ is Morita compatible if A τ G B τ G whenever A B equivariantly (roughly speaking). σ τ if the natural transformations factor this way: A max G A τ G A σ G A red G 11 / 58

12 Crossed Product Functors max is Morita compatible and exact, and red max. red is Morita compatible, but not (in general) exact. Theorem. For any second countable locally compact group G, there exists a unique minimal exact and Morita compatible crossed product E. Conjecture: For any G-C -algebra A, the E-assembly map is an isomorphism. µ E : K top (G; A) K (A E G) 12 / 58

13 Coaction Functors A coaction functor is a functor (A, δ) (A τ, δ τ ) on Ĝ-C, together with surjective natural transformations (A m, δ m ) (A τ, δ τ ) (A n, δ n ), where (A, δ) (A m, δ m ) is the maximalization functor, and (A, δ) (A n, δ n ) is the normalization functor. By the way, and A m δ m G max G = A m K, A n δ n G red G = A n K. 13 / 58

14 Coaction Functors τ is exact if 0 (I τ, γ τ ) (A τ, δ τ ) (B τ, ɛ τ ) 0 is short exact in Ĝ-C whenever 0 (I, γ) (A, δ) (B, ɛ) 0 is short exact. τ is Morita compatible if (A m, δ m ) (B m, ɛ m ) descends to (A τ, δ τ ) (B τ, ɛ τ ) whenever (A, δ) (B, ɛ) (roughly speaking). σ τ if the natural transformations factor this way: (A m, δ m ) (A τ, δ τ ) (A σ, δ σ ) (A n, δ n ). 14 / 58

15 Coaction Functors Maximalization is Morita compatible and exact, and dominates normalization. Normalization is Morita compatible, but not (in general) exact. Theorem. There exists a unique minimal exact and Morita compatible coaction functor. 15 / 58

16 Coaction Functors Theorem. For any coaction functor τ, the composition σ defined by σ : (A, α) (A max G, ˆα) ((A max G) τ, (ˆα) τ ) (A max G) τ (with the associated natural transformations) is a crossed product functor. If τ is exact, then σ will be exact; If τ is Morita compatible, then σ will be Morita compatible; If τ τ, then σ σ. If τ = τ E, we write (A max G) τ = A E G. 16 / 58

17 Abstract We use functors on a category of coactions to understand the crossed-product functors that have been recently introduced by Baum, Guentner and Willett in relation to the Baum-Connes conjecture. The most important coaction functors (to us) are the ones induced by large ideals of a Fourier-Stieltjes algebra B(G). The intersection of two large ideals of B(G) for which the associated coaction functors (and hence the associated crossed-product functors) are exact, is also exact. 17 / 58

18 Coaction Functors and Exact Large Ideals of Fourier-Stieltjes Algebras S. Kaliszewski*, Arizona State University Magnus Landstad, NTNU John Quigg, ASU AMS Fall Southeastern Sectional Meeting Special Session on the Analysis, Geometry, and Topology of Groupoids University of Memphis, TN October 17, / 58