Representation theory and (co)homology for subfactors, -lattices and - - PowerPoint PPT Presentation

Representation theory and (co)homology for subfactors, λ-lattices and C∗-tensor categories

Abel Symposium, 7-11 August 2015 Stefaan Vaes∗ (joint work with Sorin Popa and Dimitri Shlyakhtenko)

∗ Supported by ERC Consolidator Grant 614195

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A very short introduction to subfactors

Goal: representation theory and (co)homology for the standard invariant

• f subfactors.

Let N ⊂ M be a finite index subfactor.

◮ Jones projection eN : L2(M) → L2(N). ◮ Basic construction M1 = M, eN, with canonical tracial state τ. ◮ Jones tower N ⊂ M ⊂ M1 ⊂ M2 ⊂ · · · . ◮ C is the category of all M-bimodules that appear in some ML2(Mn)M. ◮ Extremality : these bimodules have equal left and right dimension. ◮ Standard invariant : the λ-lattice of multimatrix algebras M′ k ∩ Mn,

k ≤ n.

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Popa’s symmetric enveloping algebra

Extremal finite index subfactor N ⊂ M a crossed product like inclusion T ⊂ S.

◮ T = M ⊗ Mop. ◮ S = M ⊠eN Mop is the unique II1 factor

• generated by commuting copies of M and Mop,
• and a projection eN,
• that is the Jones projection for both N ⊂ M and Nop ⊂ Mop.

◮ We have TL2(S)T ∼

=

• α∈Irr C

M ⊗ Mop(Hα ⊗ Hα)M ⊗ Mop.

Recall : C is the category of M-bimodules appearing in some L2(Mn). Terminology : the SE-inclusion of N ⊂ M.

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SE-correspondences

Definition (Popa-V, 2014) An SE-correspondence of N ⊂ M is an S-bimodule that is generated by T-central vectors. Here : T ⊂ S is the SE-inclusion of N ⊂ M.

◮ Trivial SE-correspondence : L2(S). ◮ Regular (coarse) SE-correspondence : L2(S) ⊗T L2(S). ◮ Weak containment, amenability, Haagerup property, property (T) now

have “obvious” definitions. Questions : Does all this only depend on the standard invariant ? What about basic examples ?

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Positive definite functions : cp SE-multipliers

Analogy : T ⊂ S = T ⋊ Γ. A one-to-one correspondence between

◮ positive definite functions ϕ : Γ → C, ◮ and normal, completely positive T-bimodular maps ψ : S → S.

Explicitly : ψ(aug) = ϕ(g) aug. Definition (Popa-V, 2014) A cp SE-multiplier of N ⊂ M is a normal, completely positive T-bimodule map ψ : S → S.

◮ Similarly : cb SE-multipliers, CMAP, weak amenability. ◮ Since L2(S) ∼

=

α∈Irr C(Hα ⊗ Hα) as T-bimodules,

a T-bimodular map ψ : S → S must be given by multiplication with a scalar ϕ(α) on each Hα ⊗ Hα.

◮ Popa-V : an intrinsic description of positive definiteness, on arbitrary

rigid C∗-tensor categories.

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Universal C∗-algebra of a rigid C∗-tensor category

Fusion ∗-algebra C[C] with Irr(C) as vector space basis and product given by the fusion rules.

◮ Every cp-multiplier ψ : Irr(C) → C defines a positive functional

ωψ : C[C] → C : ωψ(α) = d(α) ψ(α).

◮ Not every positive functional on C[C] arises like this. ◮ We have singled out the class of admissible representations of C[C]. ◮ We can define the universal C∗-algebra Cu(C).

Obs : the ∗-algebra C[C] need not have a universal enveloping C∗-algebra. Theorem (Popa-V, 2014) There is a natural bijection between SE-correspondences of N ⊂ M and representations of the C∗-algebra Cu(C).

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Examples

Temperley-Lieb-Jones

◮ The smallest possible λ-lattice: generated by the Jones projections of

a subfactor of index λ−1.

◮ As a tensor category C = Rep(PSUq(2)) with q + 1/q = λ−1/2. ◮ Popa-V : Haagerup property and CMAP hold ; irreducible

representations labeled by [0, λ−1]. Examples with property (T)

◮ Popa-V : the tensor category Rep(SUq(3)) with 0 < q < 1 has

property (T).

◮ Gives rise to the first subfactors with a property (T) standard

invariant, not constructed from property (T) groups. Behind all this : compact quantum groups and the results of De Commer - Freslon - Yamashita (2013) and Arano (2014).

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Homology and cohomology

SE-inclusions T ⊂ S are examples of irreducible quasi-regular inclusions

• f II1 factors.

◮ The normalizer NS(T) consists of all unitaries u ∈ S with uTu∗ = T. ◮ The quasi-normalizer QNS(T) consists of all x ∈ S such that TxT

has finite index as a T-bimodule. Note that : T ⊂ S is quasi-regular, i.e. QNS(T)′′ = S. Rest of the talk (joint work with S. Popa and D. Shlyakhtenko)

◮ (Co)homology for irreducible, quasi-regular inclusions. ◮ In particular, SE-inclusions. Thus: for rigid C∗-tensor categories. ◮ L2-Betti numbers in all these cases.

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Cohomology for quasi-regular inclusions

Definition (PSV, 2015) Let T ⊂ S be an irreducible quasi-regular inclusion and put S = QNS(T). For an S-bimodule H, define Hn(T ⊂ S, H) as the cohomology of HT

d0

→ MorT(S, H)

d1

→ MorT(S ⊗T S, H)

d2

→ · · · where e.g. (d1c)(x ⊗ y) = x · c(y) − c(xy) + c(x) · y.

◮ Regular representation : the S-bimodule L2(S) ⊗T L2(S). ◮ Define M := EndS−S(L2(S) ⊗T L2(S)) with state µ given by 1 ⊗ 1. ◮ The state µ on M is faithful. It is a trace iff the T-bimodules inside

L2(S) have equal left and right dimension. Definition - for unimodular inclusions (PSV, 2015) β(2)

n (T ⊂ S) = dimM Hn(T ⊂ S, L2(S) ⊗T L2(S)).

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Further remarks

◮ A slight variant works for arbitrary quasi-regular inclusions T ⊂ S.

In particular, for Cartan inclusions : we recover Gaboriau’s L2-Betti numbers of the associated equivalence relation.

◮ The SE-inclusion of an extremal subfactor is unimodular, and we can

thus consider its L2-Betti numbers.

◮ Same remark for the SE-inclusion of any tensor category C of finite

index M-bimodules with equal left and right dimension.

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Homology for quasi-regular inclusions

◮ Similar bar complex to define homology of a quasi-regular inclusion. ◮ But : how to compute (co)homology for SE-inclusions T ⊂ S ?

Tool : Ocneanu’s tube algebra A with augmentation ǫ : A → C. Note : natural projection p0 ∈ A with p0Ap0 = C[C]. Theorem (PSV, 2015) Let C be a tensor category of M-bimodules, with Ocneanu’s tube algebra A with augmentation ǫ : A → C, and SE-inclusion T ⊂ S. Then, the Hochschild homology of an A-module K is canonically isomorphic to Hn(T ⊂ S, H) for the associated S-bimodule H. Intrinsic definition of β(2)

n (C) for any rigid C∗-tensor category C.

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Computations and further results

◮ For Temperley-Lieb-Jones, all L2-Betti numbers vanish. ◮ Expected formulae for free products, direct products :

non-vanishing β(2)

1

for Fuss-Catalan.

◮ One-cocycles can be exponentiated into cp multipliers.

Characterizations of the Haagerup property, property (T), etc.

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