Ozawas class S for locally compact groups and unique prime - - PowerPoint PPT Presentation

▶
ozawa s class s for locally compact groups and unique
SMART_READER_LITE
LIVE PREVIEW

Ozawas class S for locally compact groups and unique prime - - PowerPoint PPT Presentation

Sep 17, 2023 112 likes •468 views

Ozawas class S for locally compact groups and unique prime factorization of group von Neumann algebras Tobe Deprez Skyline Communications IPAM, Lake arrowhead, 2019 Tobe Deprez Class S for locally compact groups 1 / 27 Group von Neumann

slide-1
SLIDE 1

Ozawa’s class S for locally compact groups and unique prime factorization of group von Neumann algebras

Tobe Deprez

Skyline Communications

IPAM, Lake arrowhead, 2019

Tobe Deprez Class S for locally compact groups 1 / 27

slide-2
SLIDE 2

Group von Neumann algebra

Group G Group von Neumann algebra L(G) ◮ Consider left-regular representation λ : G → B

L2(G)

  • (λgξ)(h) = ξ(g−1h)

g, h ∈ Γ, ξ ∈ L2(G) Group algebra CG = span{λg}g∈G Definition The group von Neumann algebra L(G) is the von Neumann algebra generated by CG, i.e. L(G) = CG

w.o. = span{λg}g∈G w.o.

Tobe Deprez Class S for locally compact groups 2 / 27

slide-3
SLIDE 3

Problem setting

Group G Group von Neumann algebra L(G) Question How much does L(G) “remember” of the structure of G? ◮ (Connes, 1976) All L(G) are isomorphic for G countable, amenable, icc ◮ Open problem: is L(Fn) ∼ = L(Fm) if n = m? ◮ Ozawa’s class S

◮ G countable: (Ozawa, 2004), (Ozawa-Popa, 2004), ... ◮ G locally compact: this talk

Tobe Deprez Class S for locally compact groups 3 / 27

slide-4
SLIDE 4

Contents

1 Class S for countable groups

Definition Applications

2 Topological amenability 3 Class S for locally compact groups

Definition My results

Tobe Deprez Class S for locally compact groups 4 / 27

slide-5
SLIDE 5

Class S for countable groups Definition

Contents

1 Class S for countable groups

Definition Applications

2 Topological amenability 3 Class S for locally compact groups

Definition My results

Tobe Deprez Class S for locally compact groups 5 / 27

slide-6
SLIDE 6

Class S for countable groups Definition

Class S for countable groups

Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying η(gkh) − g · η(k) → 0 if k → ∞ Examples ◮ Free groups Fn η(k) = unif. measure on path e to k

◮ Right invariance η(kh) = unif. measure path e to kh η(k) = unif. measure path e to k difference: path from k to kh

k k kh kh

Tobe Deprez Class S for locally compact groups 6 / 27

slide-7
SLIDE 7

Class S for countable groups Definition

Class S for countable groups

Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying η(gkh) − g · η(k) → 0 if k → ∞ Examples ◮ Free groups Fn η(k) = unif. measure on path e to k

◮ Left equivariance η(gk) = unif. measure path e to gk g · η(k) = unif. measure path g to gk difference: path from e to g

g gk gk

Tobe Deprez Class S for locally compact groups 6 / 27

slide-8
SLIDE 8

Class S for countable groups Definition

Class S for countable groups

Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying η(gkh) − g · η(k) → 0 if k → ∞ Examples ◮ Free groups Fn ◮ Amenable groups

◮ ∃ sequence µn ∈ Prob(Γ) µn − g · µn → 0 ◮ Define η(k) = 1 |k|

|2k|

  • i=|k|+1

µi

e g gk gkh

Tobe Deprez Class S for locally compact groups 6 / 27

slide-9
SLIDE 9

Class S for countable groups Definition

Class S for countable groups

Γ countable group Definition (Ozawa, 2006) Γ is in class S (or is bi-exact) if Γ is exact and ∃ map η : Γ → Prob(Γ) satisfying η(gkh) − g · η(k) → 0 if k → ∞ Examples ◮ Free groups Fn ◮ Amenable groups ◮ (Adams, 1994) Hyperbolic groups ◮ (Skandalis, 1988) Lattices in finite-center, connected, simple Lie groups with real rank 1

Tobe Deprez Class S for locally compact groups 6 / 27

slide-10
SLIDE 10

Class S for countable groups Definition

Exactness

Definition (Kirchberg-Wasserman, 1999) Γ is exact if for every short exact sequence 0 → A → B → C → 0

  • f Γ-C∗-algebras, also

0 → A ⋊r Γ → B ⋊r Γ → C ⋊r Γ → 0 is exact. Examples ◮ Almost every group is exact

e.g. amenable groups, hyperbolic groups, linear groups, countable subgroups of connected simple Lie groups, ...

◮ Examples of non-exact groups: (Gromov, 2003) and (Osajda, 2014)

Tobe Deprez Class S for locally compact groups 7 / 27

slide-11
SLIDE 11

Class S for countable groups Applications

Contents

1 Class S for countable groups

Definition Applications

2 Topological amenability 3 Class S for locally compact groups

Definition My results

Tobe Deprez Class S for locally compact groups 8 / 27

slide-12
SLIDE 12

Class S for countable groups Applications

Applications

Group G Group von Neumann algebra L(G) Theorem (Ozawa, 2004) L(Γ) is solid if Γ is in class S, i.e. for every diffuse N ⊆ L(Γ) von Neumann subalgebra, the algebra N′ ∩ L(Γ) is amenable. Corollary L(Γ) is prime if Γ is non-amenable, icc and in class S, i.e. L(Γ) ∼ = M1 ⊗ M2 if M1, M2 non-type I factors. L(F2 × F2) = L(F2) ⊗ L(F2) ∼ = L(F2).

Tobe Deprez Class S for locally compact groups 9 / 27

slide-13
SLIDE 13

Class S for countable groups Applications

Applications

Group G Group von Neumann algebra L(G) Theorem (Ozawa-Popa, 2004) Let Γ = Γ1 × · · · × Γn with Γi non-amenable, icc and in class S. Then L(Γ) = L(Γ1) ⊗ . . . ⊗ L(Γn) has unique prime factorization (UPF), i.e. if L(Γ) = N1 ⊗ . . . ⊗ Nm for prime factors N1, . . . , Nm, then n = m and Ni ∼ =s L(Γi) (after relabeling). L(F2 × F2 × F2) ∼ = L(F2 × F2).

Tobe Deprez Class S for locally compact groups 10 / 27

slide-14
SLIDE 14

Topological amenability

Contents

1 Class S for countable groups

Definition Applications

2 Topological amenability 3 Class S for locally compact groups

Definition My results

Tobe Deprez Class S for locally compact groups 11 / 27

slide-15
SLIDE 15

Topological amenability

Topological amenability – Definition

◮ G locally compact and second countable ◮ X compact topological space, G X continuous Definition (Anantharaman-Delaroche, 1987) G X is (topologically) amenable if ∃ weakly* continuous maps µn : X → Prob(G) such that µn(g · x) − g · µn(x) → 0 uniformly on X and on compact sets for g ∈ G. Examples ◮ If X = {x0}, then G X is amenable iff G is amenable ◮ If X discrete and G X free, then G X amenable

µn(x) = δx

Tobe Deprez Class S for locally compact groups 12 / 27

slide-16
SLIDE 16

Topological amenability

Topological amenability – Example

Definition (Anantharaman-Delaroche, 1987) G X is (topologically) amenable if ∃ : µn : X → Prob(G) of continuous maps such that µn(g · x) − g · µ(x) → 0 uniformly on X and on compact sets for g ∈ G. Examples ◮ F2 boundary of Cayley graph µn(x) = unif. measure on first n vertices of path e to x

◮ µn(g · x) = (...) path e to g · x ◮ g · µn(x) = (...) path g to g · x difference: path from e to g

g x g · x g · x

Tobe Deprez Class S for locally compact groups 13 / 27

slide-17
SLIDE 17

Topological amenability

Topological amenability – Example

Definition (Anantharaman-Delaroche, 1987) G X is (topologically) amenable if ∃ : µn : X → Prob(G) of continuous maps such that µn(g · x) − g · µ(x) → 0 uniformly on X and on compact sets for g ∈ G. Examples ◮ F2 boundary of Cayley graph ◮ Γ boundary of Cayley graph for Γ hyperbolic

Tobe Deprez Class S for locally compact groups 13 / 27

slide-18
SLIDE 18

Topological amenability

Characterization of class S

Theorem (Ozawa, 2006) A countable group Γ belongs to class S if and only if Γ has amenable action on a boundary that is small at infinity, i.e. ∃ compactification hΓ

  • f Γ such that

◮ Actions by left and right translation extend to actions on hΓ, ◮ Action by right translation is trivial on νΓ = hΓ \ Γ, ◮ Action by left translation on νΓ = hΓ \ Γ is topologically amenable.

Tobe Deprez Class S for locally compact groups 14 / 27

slide-19
SLIDE 19

Topological amenability

Link with C∗-algebras

Consider the following conditions: (i) G X is amenable (ii) C(X) ⋊ G ∼ = C(X) ⋊r G (iii) C(X) ⋊r G is nuclear Theorem (Anantharaman-Delaroche, 1987) For G countable, we have (i) ⇔ (ii) ⇔ (iii) Theorem (Anantharaman-Delaroche, 2002) For G locally compact, we have (i) ⇒ (ii) ⇒ (iii)

Tobe Deprez Class S for locally compact groups 15 / 27

slide-20
SLIDE 20

Topological amenability

Exactness and topological amenability

Definition A group G is called exact if the operation of taking the reduced crossed product preserves short exact sequences. Consider the following conditions (i) G is exact, (ii) G βluG is amenable, (iii) C∗

r (G) is exact (i.e. taking minimal tensor product preserves

exactness) Definition Left-equivariant Stone-Čech compactification βluG G K βluG

G-equiv f i ∃!G-equiv βf

C(βluG) ∼ = C lu

b (G)

=

  • f ∈ Cb(G)
  • λgf − f ∞ → 0 if g → e
  • Tobe Deprez

Class S for locally compact groups 16 / 27

slide-21
SLIDE 21

Topological amenability

Exactness and topological amenability

Consider the following conditions (i) G is exact, (ii) G βluG is amenable, (iii) C∗

r (G) is exact (i.e. taking minimal tensor product preserves

exactness) Theorem (Kirchberg-Wasserman, 1999; Ozawa, 2000) For G countable, we have (i) ⇔ (ii) ⇔ (iii) Theorem (Anantharaman-Delaroche, 2002; Brodzki-Cave-Li, 2017) For G locally compact, we have (i) ⇔ (ii) ⇒ (iii). ◮ Remark: for locally compact (iii) ⇒ (i) is open.

Tobe Deprez Class S for locally compact groups 17 / 27

slide-22
SLIDE 22

Class S for locally compact groups Definition

Contents

1 Class S for countable groups

Definition Applications

2 Topological amenability 3 Class S for locally compact groups

Definition My results

Tobe Deprez Class S for locally compact groups 18 / 27

slide-23
SLIDE 23

Class S for locally compact groups Definition

Ozawa’s class S for locally compact groups

Definition (Brothier-D-Vaes, 2018) A locally compact group G is in class S if G is exact and ∃ continuous map η : G → Prob(G) satisfying lim

k→∞ η(gkh) − g · η(k) = 0

uniformly on compact sets for g, h ∈ G. Examples ◮ Amenable groups ◮ (Skandalis, 1988) Finite-center, connected, simple Lie groups of real rank 1

e.g. SL2(n, R), SO(n, 1), SU(n, 1), Sp(n, 1)

◮ (Brothier-D-Vaes, 2018) Automorphism groups of trees and hyperbolic graphs

Tobe Deprez Class S for locally compact groups 19 / 27

slide-24
SLIDE 24

Class S for locally compact groups My results

Contents

1 Class S for countable groups

Definition Applications

2 Topological amenability 3 Class S for locally compact groups

Definition My results

Tobe Deprez Class S for locally compact groups 20 / 27

slide-25
SLIDE 25

Class S for locally compact groups My results

Applications

Theorem (Brothier-D-Vaes, 2018) Let G be in class S, then L(G) is solid, i.e. for every diffuse N ⊆ L(G) with expectation, we have N′ ∩ L(G) is amenable. Corollary L(G) is prime if G is in class S and L(G) non-amenable factor Theorem (D, 2019) Let G = G1 × · · · × Gn with Gi locally compact groups in class S and L(Gi) nonamenable factor. Then, L(G) ∼ = L(G1) ⊗ · · · ⊗ L(Gn) has unique prime factorization, i.e. if L(G) ∼ = N1 ⊗ · · · ⊗ Nm with Ni prime, then n = m and L(Gi) ∼ =s Ni (after relabeling).

Tobe Deprez Class S for locally compact groups 21 / 27

slide-26
SLIDE 26

Class S for locally compact groups My results

Examples

Example (Suzuki) ◮ Z2 = Z/2Z acts on F2 by flipping generators K =

n∈N Z2 acts on H = ∗n∈N F2

◮ G = H ⋊ K is in class S and L(G) is nonamenable factor (Suzuki, 2016) Corollary L(G) ∼ = L(G × G) ∼ = L(G × G × G)

Tobe Deprez Class S for locally compact groups 22 / 27

slide-27
SLIDE 27

Class S for locally compact groups My results

Proof of unique prime factorization

Theorem (D) Let G = G1 × · · · × Gn with Gi in class S such that L(Gi) is a nonamenable factor. Then, L(G) ∼ = L(G1) ⊗ . . . ⊗ L(Gn) has UPF. ◮ Follows from combining

◮ UPF results from (Houdayer and Isono, 2017) and (Ando, Haagerup, Houdayer, and Marrakchi, 2018) ◮ Locally compact version of characterization of class S

Theorem (D, 2019) A locally compact group G belongs to class S if and only if it has amenable action on a compactification that is small at infinity, i.e. ∃ compactification huG of G such that ◮ Actions by left and right translation extend to actions on huG, ◮ Action by right translation is trivial on huG \ G, ◮ Action by left translation on huG is topologically amenable.

Tobe Deprez Class S for locally compact groups 23 / 27

slide-28
SLIDE 28

Class S for locally compact groups My results

UPF results from (Houdayer-Isono, 2017)

Theorem (Houdayer-Isono, 2017; Ando, Haagerup, Houdayer, and Marrakchi, 2018) A von Neumann algebra M = M1 ⊗ . . . ⊗ Mn has unique prime factorization if each Mi is a nonamenable factor satisfying strong condition (AO). Definition (Houdayer-Isono, 2017) A von Neumann algebra M with standard representation (M, H, J, P) satisfies strong condition (AO) if there exist C∗-algebras A ⊆ M and C ⊆ B(H) such that (i) A is exact and w.o. dense in M, (ii) C is nuclear and contains A, (iii) [C, JAJ] ⊆ K(H)

Tobe Deprez Class S for locally compact groups 24 / 27

slide-29
SLIDE 29

Class S for locally compact groups My results

Proof of unique prime factorization

Theorem (D) Let G = G1 × · · · × Gn with Gi in class S such that L(Gi) is a nonamenable factor. Then, L(G) ∼ = L(G1) ⊗ . . . ⊗ L(Gn) has UPF. Proof: STP: Each L(Gi) satisfies strong condition (AO) ◮ A = C∗

r (Gi) is exact and w.o. dense in M = L(Gi)

◮ C =?

◮ Gi huGi is topologically amenable C(huGi) ⋊r Gi ∼ = C(huGi) ⋊ Gi is nuclear ◮ Consider π : C(huGi) ⋊ Gi → L2(Gi) induced by covariant rep. g → λg, f → f |Gi for f ∈ C(huGi), g ∈ Gi ◮ C = π(C(huG) ⋊ Gi)

◮ [C, JAJ] ⊆ K(H)

Tobe Deprez Class S for locally compact groups 25 / 27

slide-30
SLIDE 30

Class S for locally compact groups My results

New examples

Theorem (D, 2019) Locally compact wreath products B ≀A

X H are in class S if B is amenable, H

in class S and H X such that StabH(x) is amenable for all x ∈ X. ◮ (Ozawa, 2006) same result for discrete groups Theorem (D, 2019) Class S is closed under measure equivalence ◮ (Sako, 2009) same result for discrete groups

Tobe Deprez Class S for locally compact groups 26 / 27

slide-31
SLIDE 31

Class S for locally compact groups My results

Thank you for your attention!

Tobe Deprez Class S for locally compact groups 27 / 27

slide-32
SLIDE 32

References I

  • S. Adams, Boundary amenability for word hyperbolic groups

and an application to smooth dynamics of simple groups. Topology 33 (1994), 765–783.

  • C. Anantharaman-Delaroche, Systèmes dynamiques non

commutatifs et moyennabilité. Math. Ann. 279 (1987), 297–315.

  • C. Anantharaman-Delaroche, Amenability and exactness for

dynamical systems and their C∗-algebras. T. Am. Math.

  • Soc. 354 (2002), 4153–4179.
  • H. Ando, U. Haagerup, C. Houdayer, and A. Marrakchi,

Structure of bicentralizer algebras and inclusions of type III

  • factors. 2018. Preprint. arXiv: 1804.05706.

Tobe Deprez Class S for locally compact groups 1 / 4

slide-33
SLIDE 33

References II

  • J. Brodzki, C. Cave, and K. Li, Exactness of locally compact
  • groups. Adv. Math. 312 (2017), 209–233.
  • A. Brothier, T. Deprez, and S. Vaes, Rigidity for von

Neumann algebras given by locally compact groups and their crossed products. Commun. Math. Phys. 361 (2018), 85–125.

  • A. Connes, Classification of injective factors. Cases II1, II∞,

IIIλ, λ = 1. Ann. Math. 104 (1976), 73–115.

  • S. Deprez and K. Li, Permanence properties of property A

and coarse embeddability for locally compact groups. 2014.

  • Preprint. arXiv: 1403.7111.
  • T. Deprez, Ozawa’s class S for locally compact groups and

unique prime factorization of group von Neumann algebras. To appear in P. Roy. Soc. Edinb. A.

Tobe Deprez Class S for locally compact groups 2 / 4

slide-34
SLIDE 34

References III

  • M. Gromov, Random walk in random groups. Geom. Funct.
  • Anal. 13 (2003), 73–146.
  • C. Houdayer and Y. Isono, Unique prime factorization and

bicentralizer problem for a class of type III factors. Adv.

  • Math. 305 (2017), 402–455.
  • E. Kirchberg and S. Wassermann, Exact groups and

continuous bundles of C∗-algebras. Math. Ann. 315 (1999), 169.

  • D. Osajda, Small cancellation labellings of some infinite

graphs and applications. 2014. Preprint. arXiv: 1406.5015.

  • N. Ozawa, Amenable actions and exactness for discrete
  • groups. C. R. Acad. Sci. I-Math. 330 (2000), 691–695.

Tobe Deprez Class S for locally compact groups 3 / 4

slide-35
SLIDE 35

References IV

  • N. Ozawa, Solid von Neumann algebras. Acta Math. 192

(2004), 111–117.

  • N. Ozawa, A Kurosh-type theorem for type II1 factors. Int.
  • Math. Res. Notices (2006), 1–21.
  • N. Ozawa and S. Popa, Some prime factorization results for

type II1 factors. Invent. Math. 156 (2004), 223–234.

  • H. Sako, The class S as an ME invariant. Int. Math. Res.

Notices (2009), 2749–2759.

  • G. Skandalis, Une notion de nucléarité en K-théorie (d’après
  • J. Cuntz). K-Theory 1 (1988), 549–573.
  • Y. Suzuki, Elementary constructions of non-discrete

C∗-simple groups. Proc. Am. Math. Soc. 145 (2016), 1369–1371.

Tobe Deprez Class S for locally compact groups 4 / 4