Exotic Crossed Products

Transcription

1 Exotic Crossed Products Rufus Willett (joint with Alcides Buss and Siegfried Echterhoff, and with Paul Baum and Erik Guentner) University of Hawai i WCOAS, University of Denver, November / 15

2 G : locally compact group (maybe discrete). 2 / 15

3 G : locally compact group (maybe discrete). π : G Ñ UpHq: (strongly continuous) unitary representation. 2 / 15

4 G : locally compact group (maybe discrete). π : G Ñ UpHq: (strongly continuous) unitary representation. Definition (Brown-Guentner, 2 / 15

5 G : locally compact group (maybe discrete). π : G Ñ UpHq: (strongly continuous) unitary representation. Definition (Brown-Guentner, but Kunze-Stein 1960, Gelfand-Naimark 1940, Mehler 1880,...) For p P r2, 8q, pπ, Hq is an L p -representation if the set is dense. tpη, ξq P H H g ÞÑ x η, πpgqξ y is in L p pgqu 2 / 15

6 G : locally compact group (maybe discrete). π : G Ñ UpHq: (strongly continuous) unitary representation. Definition (Brown-Guentner, but Kunze-Stein 1960, Gelfand-Naimark 1940, Mehler 1880,...) For p P r2, 8q, pπ, Hq is an L p -representation if the set is dense. tpη, ξq P H H g ÞÑ x η, πpgqξ y is in L p pgqu Examples: The regular representation on L 2 pgq is an L p -representation for all p. 2 / 15

7 G : locally compact group (maybe discrete). π : G Ñ UpHq: (strongly continuous) unitary representation. Definition (Brown-Guentner, but Kunze-Stein 1960, Gelfand-Naimark 1940, Mehler 1880,...) For p P r2, 8q, pπ, Hq is an L p -representation if the set is dense. tpη, ξq P H H g ÞÑ x η, πpgqξ y is in L p pgqu Examples: The regular representation on L 2 pgq is an L p -representation for all p. The trivial representation on C is an L p -representation only for p 8. 2 / 15

8 G : locally compact group (maybe discrete). π : G Ñ UpHq: (strongly continuous) unitary representation. Definition (Brown-Guentner, but Kunze-Stein 1960, Gelfand-Naimark 1940, Mehler 1880,...) For p P r2, 8q, pπ, Hq is an L p -representation if the set is dense. tpη, ξq P H H g ÞÑ x η, πpgqξ y is in L p pgqu Examples: The regular representation on L 2 pgq is an L p -representation for all p. The trivial representation on C is an L p -representation only for p 8. If p ą q, then any L q -representation is an L p -representation. 2 / 15

9 1 L p -representations 2 Crossed product functors 3 Exactness of the Baum-Connes conjecture 3 / 15

10 L p -representations L p -representation of G: has dense set of matrix coefficients in L p pgq. Examples? 4 / 15

11 L p -representations L p -representation of G: has dense set of matrix coefficients in L p pgq. Examples? φ : G Ñ C is positive type if for tg 1,..., g n u Ď G, tz 1,..., z n u Ď C, nÿ i,j 1 z i z j φpg 1 i g j q ě 0. 4 / 15

12 L p -representations L p -representation of G: has dense set of matrix coefficients in L p pgq. Examples? φ : G Ñ C is positive type if for tg 1,..., g n u Ď G, tz 1,..., z n u Ď C, nÿ i,j 1 z i z j φpg 1 i g j q ě 0. GNS representation π φ : G Ñ UpHq, vector ξ with φpgq xξ, π φ pgqξy. 4 / 15

13 L p -representations L p -representation of G: has dense set of matrix coefficients in L p pgq. Examples? φ : G Ñ C is positive type if for tg 1,..., g n u Ď G, tz 1,..., z n u Ď C, nÿ i,j 1 z i z j φpg 1 i g j q ě 0. GNS representation π φ : G Ñ UpHq, vector ξ with φpgq xξ, π φ pgqξy. Lemma (Dixmier?) If φ P L p pgq is positive type, then π φ is an L p -representation. Proof: The dense subset C c pgq ξ gives rise to L p matrix coefficients. 4 / 15

14 L p -representations Positive type functions in L p pgq GNS L p -representations. 5 / 15

15 L p -representations Positive type functions in L p pgq GNS L p -representations. Example: G F 2. 5 / 15

16 L p -representations Positive type functions in L p pgq GNS L p -representations. Example: G F 2. Theorem (Haagerup?) For each fixed t P r0, 8s, the function φ t : F 2 Ñ C, g ÞÑ e t g is positive type. 5 / 15

17 L p -representations As for n ě 2, tg P F 2 g nu 4 3 n 1 6 / 15

18 L p -representations As for n ě 2, we have tg P F 2 g nu 4 3 n 1 8ÿ 8ÿ }φ t } p l p pgq 1 ` p4 3 n 1 qpe tn q p pe tp 3q n, n 1 n 1 6 / 15

19 L p -representations As for n ě 2, tg P F 2 g nu 4 3 n 1 we have 8ÿ 8ÿ }φ t } p l p pgq 1 ` p4 3 n 1 qpe tn q p pe tp 3q n, which is finite if and only if p ą logp3q{t. n 1 n 1 6 / 15

20 L p -representations As for n ě 2, tg P F 2 g nu 4 3 n 1 we have 8ÿ 8ÿ }φ t } p l p pgq 1 ` p4 3 n 1 qpe tn q p pe tp 3q n, n 1 which is finite if and only if p ą logp3q{t. Extrapolate: for any p ą q ě 2, F 2 has L p -representations which are not L q -representations. n 1 6 / 15

21 L p -representations As for n ě 2, tg P F 2 g nu 4 3 n 1 we have 8ÿ 8ÿ }φ t } p l p pgq 1 ` p4 3 n 1 qpe tn q p pe tp 3q n, n 1 which is finite if and only if p ą logp3q{t. Extrapolate: for any p ą q ě 2, F 2 has L p -representations which are not L q -representations. n 1 Similar example: G SLp2, Rq: the analogue of φpgq e g is ˆa 1 0 φ k 0 a 1 k 1 ż 2π 2 2π 0 pa 2 ` a 2 q ` pa 2 a 2 q cospθq dθ. 6 / 15

22 L p -representations Definition (Brown-Guentner) C p pgq is the completion of C c pgq for the norm }f } : supt}πpf q} BpHq ph, πq an L p -representationu. 7 / 15

23 L p -representations Definition (Brown-Guentner) C p pgq is the completion of C c pgq for the norm }f } : supt}πpf q} BpHq ph, πq an L p -representationu. Remarks: For any G, C 8pGq C maxpgq and C 2 pgq C redpgq. 7 / 15

24 L p -representations Definition (Brown-Guentner) C p pgq is the completion of C c pgq for the norm }f } : supt}πpf q} BpHq ph, πq an L p -representationu. Remarks: For any G, C 8pGq C maxpgq and C 2 pgq C redpgq. The duals p G p : { C p pgq are nested, i.e. 2 ď p ď q ď 8 ñ p G 2 Ď p G p Ď p G q Ď p G 7 / 15

25 L p -representations Definition (Brown-Guentner) C p pgq is the completion of C c pgq for the norm }f } : supt}πpf q} BpHq ph, πq an L p -representationu. Remarks: For any G, C 8pGq C maxpgq and C 2 pgq C redpgq. The duals p G p : { C p pgq are nested, i.e. 2 ď p ď q ď 8 ñ p G 2 Ď p G p Ď p G q Ď p G If G is amenable, C p pgq C q pgq for all p, q (and so p G p p G q ). 7 / 15

26 L p -representations Example: G SLp2, Rq. 8 / 15

27 L p -representations Example: G SLp2, Rq. 8 / 15

28 L p -representations Example: G SLp2, Rq. Theorem (Wiersma, Stein?) C p pgq C q pgq for all 8 ě p ą q ě 2. 8 / 15

29 L p -representations Example: G SLp2, Rq. Theorem (Wiersma, Stein?) C p pgq C q pgq for all 8 ě p ą q ě 2. On the other hand, KpC p pgqq K pc q pgqq for all p, q. 8 / 15

30 L p -representations Example: G F 2. 9 / 15

31 L p -representations Example: G F 2. 9 / 15

32 L p -representations Example: G F 2. Theorem (Okayasu, Cuntz, Pimsner-Voiculescu, Buss-Echterhoff-W.) C p pgq C q pgq for all 8 ě p ą q ě 2. 9 / 15

33 L p -representations Example: G F 2. Theorem (Okayasu, Cuntz, Pimsner-Voiculescu, Buss-Echterhoff-W.) C p pgq C q pgq for all 8 ě p ą q ě 2. Moreover, for all p: K i pc p pgqq " Z i 0 Z Z i 1 9 / 15

34 Crossed product functors How to prove that for all p, q? K pc p pf 2 qq K pc q pf 2 qq 10 / 15

35 Crossed product functors How to prove that for all p, q? K pc p pf 2 qq K pc q pf 2 qq Strategy: 1 Define an exotic crossed product functor p : tg-c -algebrasu Ñ tc -algebrasu, A ÞÑ A p G such that C p G C p pgq. 10 / 15

36 Crossed product functors How to prove that for all p, q? K pc p pf 2 qq K pc q pf 2 qq Strategy: 1 Define an exotic crossed product functor p : tg-c -algebrasu Ñ tc -algebrasu, A ÞÑ A p G such that C p G C p pgq. 2 Show that it gives rise to a descent functor p : KK G Ñ KK. 10 / 15

37 Crossed product functors How to prove that for all p, q? K pc p pf 2 qq K pc q pf 2 qq Strategy: 1 Define an exotic crossed product functor p : tg-c -algebrasu Ñ tc -algebrasu, A ÞÑ A p G such that C p G C p pgq. 2 Show that it gives rise to a descent functor p : KK G Ñ KK. 3 Use that in KK G, C is equivalent to an amenable G-C -algebra A (Cuntz), and here all crossed products are the same. 10 / 15

38 Crossed product functors Definition (Baum-Guentner-W.?) A crossed product is a functor : tg-c -algebrasu Ñ tc -algebrasu, A ÞÑ A G such that there are surjective natural transformations A max G A G A red G which compose to the standard quotient A max G A red G. 11 / 15

39 Crossed product functors Definition (Baum-Guentner-W.?) A crossed product is a functor : tg-c -algebrasu Ñ tc -algebrasu, A ÞÑ A G such that there are surjective natural transformations A max G A G A red G which compose to the standard quotient A max G A red G. Examples: max, red, but many others due to Brown-Guentner, Kaliszewski-Landstad-Quigg, / 15

40 Crossed product functors Theorem (Buss-Echterhoff-W.) Let be a crossed product functor. The following are equivalent. 1 If pap is a G-invariant corner of A, then ppapq G Ñ A G is injective. 12 / 15

41 Crossed product functors Theorem (Buss-Echterhoff-W.) Let be a crossed product functor. The following are equivalent. 1 If pap is a G-invariant corner of A, then is injective. ppapq G Ñ A G 2 extends to a functor on the category of G-C -algebras and equivariant cp maps. 12 / 15

42 Crossed product functors Theorem (Buss-Echterhoff-W.) Let be a crossed product functor. The following are equivalent. 1 If pap is a G-invariant corner of A, then is injective. ppapq G Ñ A G 2 extends to a functor on the category of G-C -algebras and equivariant cp maps. 3 extends to a functor on the category of G-C -algebras and equivariant correspondences (in the sense of Rieffel, Echterhoff-Kaliszewski-Quigg-Raeburn). 12 / 15

43 Crossed product functors Theorem (Buss-Echterhoff-W.) Let be a crossed product functor. The following are equivalent. 1 If pap is a G-invariant corner of A, then is injective. ppapq G Ñ A G 2 extends to a functor on the category of G-C -algebras and equivariant cp maps. 3 extends to a functor on the category of G-C -algebras and equivariant correspondences (in the sense of Rieffel, Echterhoff-Kaliszewski-Quigg-Raeburn). If satisfies these conditions, we call it a correspondence functor. 12 / 15

44 Crossed product functors Theorem (Buss-Echterhoff-W.) Any correspondence functor induces a descent morphism : KK G Ñ KK. With a little more work, if G is K-amenable (e.g. G F 2, SLp2, Rq), then all the crossed products C -algebras have the same K-theory. ta G a correspondence functoru 13 / 15

45 Crossed product functors Theorem (Buss-Echterhoff-W.) Any correspondence functor induces a descent morphism : KK G Ñ KK. With a little more work, if G is K-amenable (e.g. G F 2, SLp2, Rq), then all the crossed products C -algebras have the same K-theory. ta G a correspondence functoru Theorem (Kaliszewski-Landstad-Quigg, Buss-Echterhoff-W. ) Define a completion of C c pg, Aq by taking the covariant pair pa, Gq Ñ pa max Gq b C p pgq, a ÞÑ a, g ÞÑ g b g, integrating to C c pg, Aq, and completing to A p G. Then p is a correspondence functor, and C p G C p pgq. 13 / 15

46 Exactness of the Baum-Connes conjecture For simplicity: G discrete. The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. µ : K top pg; Aq Ñ K pa red Gq 14 / 15

47 Exactness of the Baum-Connes conjecture For simplicity: G discrete. The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. µ : K top pg; Aq Ñ K pa red Gq Lemma If 0 Ñ I Ñ A Ñ B Ñ 0 is a short exact sequence of G-C -algebras and Baum-Connes holds, then is exact. K i pi q Ñ K i paq Ñ K i pbq 14 / 15

48 Exactness of the Baum-Connes conjecture For simplicity: G discrete. The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. µ : K top pg; Aq Ñ K pa red Gq Lemma If 0 Ñ I Ñ A Ñ B Ñ 0 is a short exact sequence of G-C -algebras and Baum-Connes holds, then is exact. K i pi q Ñ K i paq Ñ K i pbq Theorem (Higson-Lafforgue-Skandalis) For certain non-exact groups, this fails. 14 / 15

49 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? 15 / 15

50 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? Theorem (Baum-Guentner-W., Buss-Echterhoff-W.) Define a crossed product e by completing C c pg, Aq in the representation pa, Gq Ñ pa b l 8 pgqq max G, a ÞÑ a b 1, g ÞÑ g 1 e is a correspondence functor. 15 / 15

51 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? Theorem (Baum-Guentner-W., Buss-Echterhoff-W.) Define a crossed product e by completing C c pg, Aq in the representation pa, Gq Ñ pa b l 8 pgqq max G, a ÞÑ a b 1, g ÞÑ g 1 e is a correspondence functor. 2 e takes short exact sequences to short exact sequences. 15 / 15

52 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? Theorem (Baum-Guentner-W., Buss-Echterhoff-W.) Define a crossed product e by completing C c pg, Aq in the representation pa, Gq Ñ pa b l 8 pgqq max G, a ÞÑ a b 1, g ÞÑ g 1 e is a correspondence functor. 2 e takes short exact sequences to short exact sequences. 3 e red if and only if G is exact. 15 / 15

53 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? Theorem (Baum-Guentner-W., Buss-Echterhoff-W.) Define a crossed product e by completing C c pg, Aq in the representation pa, Gq Ñ pa b l 8 pgqq max G, a ÞÑ a b 1, g ÞÑ g 1 e is a correspondence functor. 2 e takes short exact sequences to short exact sequences. 3 e red if and only if G is exact. 4 If G is non-amenable, e ă max. 15 / 15

54 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? Theorem (Baum-Guentner-W., Buss-Echterhoff-W.) Define a crossed product e by completing C c pg, Aq in the representation pa, Gq Ñ pa b l 8 pgqq max G, a ÞÑ a b 1, g ÞÑ g 1 e is a correspondence functor. 2 e takes short exact sequences to short exact sequences. 3 e red if and only if G is exact. 4 If G is non-amenable, e ă max. 5 None of the previous Baum-Connes counterexamples apply to the reformulated conjecture using e. 15 / 15

55 Exactness of the Baum-Connes conjecture Question: to what extent is exactness the only obstruction to Baum-Connes? Theorem (Baum-Guentner-W., Buss-Echterhoff-W.) Define a crossed product e by completing C c pg, Aq in the representation pa, Gq Ñ pa b l 8 pgqq max G, a ÞÑ a b 1, g ÞÑ g 1 e is a correspondence functor. 2 e takes short exact sequences to short exact sequences. 3 e red if and only if G is exact. 4 If G is non-amenable, e ă max. 5 None of the previous Baum-Connes counterexamples apply to the reformulated conjecture using e. 6 Some previous counterexamples become confirming examples. 15 / 15