Seminarvortrag: Amenability and Property T for groups and von Neumann algebras

Thomas Schick^{∗} Uni Göttingen Germany

Last compiled October 31, 2006; last edited July 30, 2001 or later

1 Definitions via representations

1.1 Definition. Relative Property T via representations for a locally compact group G with closed subgroup H:

If every (strongly continuous) unitary representation with almost G-invariant vectors (i.e. a sequence x_{n} of unit vectors such that |π(g)x_{n} − x_{n}| ⟶^{n→∞} 0 for all g ∈ G uniformly on compact subset of G) has a non-zero strictly H-invariant vector.

1.2 Lemma. In this case, the H-invariant vector x can be chosen close to x_{n} for some large n.

On the other hand, it suffices to find a finite dimensional H-invariant subspace.

1.3 Definition. G is amenable if and only if the left regular representation has almost invariant vectors.

1.4 Definition. G has the Haagerup property (is a-T-menable) if and only if there is some strongly continous unitary representaion with almost invariant vectors, and with matrix coefficients < π(g)v, w >⟶^{g→∞} 0.

1.5 Example. The left regular representation λ_{G} has G-invariant vectors if and only if G is compact, and has almost invariant vectors if and only if G is amenable.

1.6 Lemma. (G, H) relative property T with H non-compact implies G has not the Haagerup property, which implies that G is not amenable.

1.7 Example. • Abelian, solvable groups are amenable.

• Sl_{n}(K) for n ≥ 3, Sp_{2n}(K) for n ≥ 2 and K a local field have property T, more generally the K-points of connected algebraic almost simply groups over K with rk_{K}(G) ≥ 2. (Kazhdan, Vaserstein,…).

∗e-mail: schick@uni-math.gwdg.de www: http://www.uni-math.gwdg.de/schick Fax: ++49 -551/39 2985

• SO(n, 1) and SU(n, 1) do not have property T, but Sp(n, 1) and F_{4(−20)} has (Kazhdan).

• (Z^{2} ⋊ Sl_{2}(Z), Z^{2}) and ((S^{2})^{∗} ⋊ Sl_{2}(K), (S^{2})^{∗}(K)) have relative property T.

• Free groups neither have property T nor are amenable, but are a-T-menable. Same for SO(n, 1), SU(n, 1).

For von Neumann algebras: replace unitary representations by bimodules (Connes’ correspondences).

1.8 Definition. (L, τ) a von Neumann algebra with faithful normal tracial state, Q ⊂ L a von Neumann subalgebra have relative property (T) if and only if every L-L bimodule with almost central almost tracial vectors admits a sequence of almost tracial Q-central unit vectors.

Here a bimodule is a Hilbert space H with left-right normal unital actions. (x_{n} ∈ H) is almost central, if |ax_{n} − x_{n}a| ⟶^{0} ∀a ∈ L. (x_{n}) is almost tracial, if

∥⟨x_{n}, ·x_{n}⟩ − τ∥ ⟶^{0} and ∥⟨x_{n}, x_{n}·⟩ − τ∥ ⟶^{0}. x ∈ H is central, if ax = xa for all

a ∈ L.

1.9 Definition. (L, τ) is amenable if and only if the Hilbert bimodule L^{2}(L, τ)⊗ L^{2}(L, τ) with action axb := (a⊗1)x(1⊗b) contains almost central almost tracial vectors.

1.10 Theorem. Any von Neumann algebra L generated by the representation of a (discrete?) amenable group is amenable.

Moreover, if Γ is discrete then L(Γ) amenable implies Γ amenable.

Proof. Use perhaps x_{n} ⊗ x_{n}, and reasoning similar to the one of the next theorem?

1.11 Theorem. Let H ≤ G be countable groups. Then: (G, H) has relative property T if and only if (L(G), L(H)) has relative property T.

Proof. “ ⟹ ”: Let V be a bimodule with almost central almost tracial vectors (x_{n}). Define G-action on V by gx := u_{g}xu_{g}^{∗}. Because of relative property T (strong version) exists to ϵ > 0 an n ∈ ℕ and an H-invariant vector x with |x − x_{n}| < ϵ, ∥⟨x_{n}, ·x_{n}⟩ − τ∥ < ϵ, ∥⟨x_{n}, x_{n}·⟩ − τ∥ < ϵ. These x produce an L(H)-central almost tracial sequence.

“leftarrow”: If τ: G → U(V) has normed almost invariant vectors (x_{n}), we need to find a finite dimensional H-invariant subspace. Set K := l^{2}(G)⊗V with L(G)-bimodule structure given by u_{g}(δ_{h} ⊗ x)u_{v} = δ_{ghv} ⊗ π(g)x. Then δ_{e} ⊗ x_{n} are almost central and almost tracial. Therefore we have μ ≠ 0 L(H)-invariant. Now K ≅ l^{2}(G, V), and μ(hgh^{−1}) = π(h)μ(g) for all h ∈ H, g ∈ G. If μ(g) ≠ 0, L^{2} implies that |{| hgh^{−1} | h ∈ H} < ∞. The corresponding linear span of μ(hgh^{−1})_{h∈H} is a finite dimensional H-invariant subspace of V.

1.12 Theorem. (Popa): If L^{∞}(X, μ) ⋊ G ⊃ P are von Neumann algebras with relative property T, then one can essentially conjugate P to L(G) with a unitary.

1.13 Theorem. (Popa) The functor from discrete countable ICC groups with property T (and inclusions as homomorphisms) to von Neumann algebras: G ↦ L(⊕_{g∈G}ℤ ⋊ G) is injective upon passage to isomorhpism classes.

1.14 Remark. This is a partial result towards the conjecture of Connes, which states the same for the functor G ↦ L(G).

1.15 Proposition. If L is a II_{1}-factor (really, only a faithful trace is needed) and Q ⊂ L has relative property T, and 0 ≠ p ∈ Q is a projection, then pQp ⊂ pLp also has relative property T.

Proof. Because L is a factor there are partial isometries v_{1}, …, v_{k} with v_{1} = p, v_{i}^{∗}v_{i} ≤ p and ∑ v_{i}v_{i}^{∗} = 1.

Assume that K_{1} is a pLp bimodules with almost tracial almost invariant vectors x_{n}. Set K := LpK_{1}pL with ⟨xξy^{∗}, aηb^{∗}⟩ := ⟨ξ, (x^{∗}a)η(b^{∗}y)⟩. Upon normalization, ∑_{i} v_{i}x_{n}v_{i}^{∗} gives almost central almost traical vectors for the L-L bimodule. Therefore, we get a sequence (μ_{n}) of almost tracial Q-central vectors. Upon normalization, pμ_{n} = μ_{n}p is pQp-central almost tracial.

2 Definitions via cohomology

2.1 Definition. Let π: G → U(H_{π}) be a strongly continuous representation of a locally compact group. Set

(1) Z^{1}(G, π) := {b: G ⟶^{C} H_{π} | g(gh) = π(g)b(h) + b(g)}

(2) B^{1}(G, π) = {b ∈ Z^{1} | ∃x ∈ H_{π} : b(g) = π(g)x − x}

(3) H^{1}(G, π) = Z^{1}/B^{1}

(4) \overline{H^{1}}(G, π) = Z^{1}/\overline{B}^{1}, where we use the topology of uniform convergence on compact sets.

2.2 Remark. There is an interpretation in terms of affine actions: b: G → H_{π} defines α(g): H_{π} → H_{π} with α(g)v = π(g)v + b(g). This is an affine action with linear part π if and only if b ∈ Z^{1}(G, π). Moreover, b ∈ B^{1}(G, π) if and only if α has a fixed point (if and only if α is conjugate to π by a translation). The center lemma implies that this is the case if and only if b is bounded.

We observe: H^{1}(G, π) = 0 if and only if every affine action (with linear part π) has a fixed point, and \overline{H^{1}}(G, π) = 0 if and only if every such action has almost fixed points.

2.3 Lemma. If G is locally compact σ-compact, and π has almost invariant vectors (but no invariant vectors), then H^{1}(G, ⨁_{k=1}^{∞} π) ≠ 0.

Proof. Write G = ⋃ L_{n} for compact subsets L_{n}. We find normed x_{n} with max_{g∈L_{n}} |gx_{n} − x_{n}| ≤ 2^{−n}. Set b(g) := ⨁ n(gx_{n} − x_{n}); then b converges uniformly on all compact subsets. But b is unbounded by the following reasoning: Otherwise: if C is the closed convex hull of π(G)x_{n} (for n large fixed), then C is G-invariant, |gx_{n} − x_{n}| < 1 implies Re(⟨gx_{n}, x_{n}⟩) ≥ 1/2 for all g ∈ G, so 0 ∉ C. The vector of minimal length in C is then G-invariant, contradicting our assumption on π.

2.4 Theorem. (Delorme-Guichardet): If G is locally compact and σ-compact then we have the equivalence: G has property T ⟺ H^{1}(G, π) = 0 for every unitary representation π ⟺ every affine isometric action on a Hilbert space has a fixed point.

2.5 Corollary. Every action of a σ-compact group G with property T on a tree fixed an edge or a vertex

Proof. Otherwise, construct an affine isometric action which is metrically proper, i.e. doesn’t have almost fixed points.

2.6 Theorem. (Shalom) If G is locally compact 2nd countable and compactly generated, we have the further equivalence: G has property T ⟺ \overline{H^{1}}(G, π) = 0∀π ⟺ \overline{H^{1}}(G, π) = 0 for all irreducible π ⟺ H^{1}(G, π) = 0 for all irreducible π.

2.7 Proposition. Equivalent to: G has the Haagerup property is: every continuous isometric action of G on an affine Hilbert space H is metrically proper (i.e. if B ⊂ H is bounded, then {g ∈ G | gB ∩ B ≠ ∅} ⊂ G is precompact).

3 Some properties

3.1 Theorem. f: G_{1} → G_{2} continuous with dense image.

• If G_{1} has property T, so has G_{2}

• If G_{1} has property T and G_{2} is a-T-menable, then im(f) is precompact.

• G/\overline{[G, G]} is compact, G is unimodular

Proof. For the first statement, use the restrictions G_{1} → G_{2} ⟶^{π} U(H_{π}) and denseness. The other two follow from the first.

3.2 Theorem. If G locally compact has property T, it is compactly generated.

Proof. Write G = ⋃ H over the set C all compactly generated open subgroups. If H ∈ C, then G/H is discrete.

The representation π = ⨁_{H∈C} λ_{G/H} has almost invariant vectors (since every compact subset L is contained in some H_{L} ∈ C, then δ_{H_{L}} is L-fixed). Property T yields 0 ≠ x = ⊕x_{H}. In particular, for some H, x_{H} ∈ l^{2}(G/H) is G-fixed. Because of the l^{2}-condition, G/H is finite, since H was compactly generated, therefore so is G.

3.3 Proposition. Let 1 → N → G → G/N → 1 be an exact sequence of locally compact groups, N closed in G.

If N, G/N have property T, so has G. If G has property T, so have G/N and N.

Same hold for amenability.

Proof. Use induction and restriction of representations, not trivial.

3.4 Proposition. If C ≤ G is closed and central, G/C has property T and G/\overline{[G, G]} is compact, then G has property T.

3.5 Proposition. G is amenable if and only if the canonical map C_{max}^{∗}G → C_{red}^{∗}G is an isomorphism.

4 Amenability and means

4.1 Theorem. G locally compact is amenable if and only if there is a finitely additive G-invariant mean (non-negative, μ(1) = 1) on L^{∞}(G).

4.2 Example. The Haar measure produces this if G is compact; otherwise one needs axiom of choice, even for G = ℤ.

4.3 Theorem. G is amenable if and only if any continuous affine action of G on a non-empty compact convex subset X of a locally convex topological vector space has a fixed point.

Proof. Part: the set of means in the dual of L^{∞} is such a set, an invariant mean is a fixed point for this action.

5 Amenability and von Neumann algebras

5.1 Theorem. L ⊂ B(H) is amenable if and only if there is a linear projection P: B(H) → L of norm 1. (Such a P is automatically an L-bimodule map). (This property is called “injective”, and is injectivity in an appropriate abelian category of von Neumann algebras).

This is equivalent to: L is hyperfinite (i.e. generated by an increasing sequence of finite dimensional subalgebras).

This is furthermore equivalent to: all finite derivations of L with coefficients in a normal dual Banach bimodule over L are inner (a condition on H^{1}!).

5.2 Theorem. If an action on a measure space is amenable (or more generally a groupoid is amenable) then the induced (crossed product) von Neumann algebra is amenable.

5.1 Actions

We consider action of countable G “freely” and ergodically on the standard probability space ([0, 1], dμ).

5.3 Definition. (1) Such actions are conjugate if they are transported by group isomorphisms and measure space isomorphism.

(2) They are orbit equivalence, if we have a measure space isomorphism mapping (almost all) orbits bijectively to orbits.

(3) We can also just consider the von Neumann algebra L^{∞}(X, μ) ⋊ G.

5.4 Theorem. (Connes-Feldmann-Weiss, Ornstein-Weiss): if the equivalence relation is amenable, then it is orbit equivalent to the standard action of ℤ on ∏_{g∈ℤ}(X, μ).

Here, the equivalence relation Q ⊂ X × X is amenable if for every Banach space E and cocycle m: Q → Iso(E) (m(x, y)m(y, z) = m(x, z)) and minvariant Borel field A_{x} of compact convex subsets of E^{∗} (m-invariant menas that (m(x, y)^{∗})^{−1}A_{y} = A_{x} almost everywhere), there is an m-invariant section ϕ: X → E^{∗} with ϕ(x) ∈ A_{x}, where m-invariant means (m(x, y)^{∗})^{−1}ϕ(y) = ϕ(x) almost e.

5.5 Remark. Amenable actions produce amenable equivalence relations.

5.6 Definition. An action (of a discrete group) on X is amenable if there are b^{n}: X → prob(G) such that ∀g ∈ G lim_{n→∞} sup_{x∈X} |gb_{x}^{n} − b_{gx}^{n}|_{l^{1}} = 0.

If X is compact Hausdorff, we want w^{∗}-continuity of b^{n} and talk of topological amenability, if X is a measure space: need suitable measurability (?).

5.7 Remark. G is amenable if and only if it acts amenably on a point, then every action of G is amenable.

G acts amenably on some compact space, if it acts amenably on its Stone-Cech compactification.

6 Spectral approaches to property T and amenability

6.1 Definition. Assume μ is a probability measure on G and π a unitary action. We define a Laplacian π(μ) ∈ B(H_{π}) by

⟨π(μ)x, y⟩ = ∫_{G}⟨π(g)x, y⟩ dμ(g).

Then ∥π(μ)∥ ≤ 1.

6.2 Proposition. If G is locally compact, μ is Haar absolutely continuous and the support of μ topologically generates G, then π has almost invariant vectors if and only if 1 ∈ spec(π(μ)).

If the support of μ^{∗}∗μ topologically generates G, this is equivalent to ∥π(μ)∥ = 1.

6.3 Proposition. If G is discrete, it is amenable if and only if it admits a Folner exhaustion.

6.4 Theorem. If G = π_{1}(X) (discrete) then H^{1}(G, π) = H_{dR}^{1}(X, X̃ ×_{G} H_{π}) by Hodge theory. This gives property T if X is Riemannian symmetric irreducible of non-compact type (different from H^{n}(R), H^{n}(C)), using Bochner formulas.

In a different direction (without Hodge theory):

6.5 Theorem. (Zuk): If X is a locally finite simplicial complex (simply connected) and for every vertex v ∈ X, the first (non-zero) eigenvalue of the Laplacian of the link of v is > 1/2 ((Δf)(x) = f(x) − 1/deg(x) ∑_{y∼x} f(y)), and if G acts properly and cocompactly on X, then G has property T.

References

[1] Alain Connes. Noncommutative geometry. Academic Press Inc., San Diego, CA, 1994.

[2] Nigel Higson and John Roe. Amenable group actions and the Novikov conjecture. J. Reine Angew. Math., 519:143–153, 2000.

[3] Alan L. T. Paterson. Groupoids, inverse semigroups, and their operator algebras, volume 170 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1999.

[4] Stefaan Vaes. Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa).