university of copenhagenoperator_2014/slides/3_1... · 2014-08-13 ·...

university of copenhagen

The weak Haagerup property

Søren Knudby

ICM Satellite Conference on Operator Algebras and ApplicationsCheongpung 2014

August 11, 2014

u n i v e r s i t y o f c o p e n h a g e n

Why study approximation properties?

They can be used as• invariants of groups and operator algebras.Example: C ∗r (SL(2,Z)) has the completely boundedapproximation property. C ∗r (SL(3,Z)) does not.

• a tool to prove fantastic theorems.• Cartan-rigidity• The Baum-Connes conjecture holds for groups with the

Haagerup property

Slide 2/12 — Søren Knudby — The weak Haagerup property — August 11, 2014


Why study approximation properties?They can be used as

• invariants of groups and operator algebras.

Example: C ∗r (SL(2,Z)) has the completely boundedapproximation property. C ∗r (SL(3,Z)) does not.


Haagerup property




• invariants of groups and operator algebras.Example: C ∗r (SL(2,Z)) has the completely boundedapproximation property. C ∗r (SL(3,Z)) does not.


Haagerup property





• a tool to prove fantastic theorems.

• Cartan-rigidity• The Baum-Connes conjecture holds for groups with the

Haagerup property





• a tool to prove fantastic theorems.• Cartan-rigidity

• The Baum-Connes conjecture holds for groups with theHaagerup property





• a tool to prove fantastic theorems.• Cartan-rigidity• The Baum-Connes conjecture

holds for groups with theHaagerup property






Haagerup property



DefinitionsA linear map M : A→ B between C ∗-algebras is positive ifM (A+) ⊆ B+, and it is completely positive if

M ⊗ idn : A⊗Mn(C)→ B ⊗Mn(C)

is positive for all n ∈ N.

The map M is completely boundedif

‖M‖cb = supn‖M ⊗ idn‖ <∞.

Completely positive maps are completely bounded.



DefinitionsA linear map M : A→ B between C ∗-algebras is positive ifM (A+) ⊆ B+, and it is completely positive if

M ⊗ idn : A⊗Mn(C)→ B ⊗Mn(C)

is positive for all n ∈ N. The map M is completely boundedif

‖M‖cb = supn‖M ⊗ idn‖ <∞.

Completely positive maps are completely bounded.



DefinitionsG always denotes a locally compact Hausdorff group.

λ is the left regular representation of G on L2(G), andL(G) = λ(G)′′ is the group von Neumann algebra.

Let u : G → C be continuous.Then u is positive definite, if

Mu : λ(g) 7→ u(g)λ(g) (g ∈ G)

extends to a normal, completely positive map on L(G).

u is a Herz-Schur multiplier, if

Mu : λ(g) 7→ u(g)λ(g) (g ∈ G)

extends to a normal, completely bounded map on L(G).

‖u‖cb = ‖Mu‖cb



Approximation properties for groups

ExampleG is amenableiff there is a net (ui)i∈I of continuous functions ui : G → Csuch that

• ui → 1 uniformly on compact subsets of G,• ui is compactly supported,• ui is positive definite.

The best (lowest) value of supi ‖ui‖cb is the weak Haagerupconstant of G, denoted ΛWH(G).

ΛWH(G) ∈ [1,∞] and

ΛWH(G) <∞ ⇐⇒ G has the weak Haagerup property.




ExampleG is weakly amenableiff there is a net (ui)i∈I of continuous functions ui : G → Csuch that

• ui → 1 uniformly on compact subsets of G,• ui is compactly supported,• ui is a Herz-Schur multiplier, and supi ‖ui‖cb <∞.


ΛWH(G) ∈ [1,∞] and





ExampleG has the Haagerup propertyiff there is a net (ui)i∈I of continuous functions ui : G → Csuch that

• ui → 1 uniformly on compact subsets of G,• ui vanishes at infinity,• ui is positive definite.


ΛWH(G) ∈ [1,∞] and





ExampleG has the weak Haagerup propertyiff there is a net (ui)i∈I of continuous functions ui : G → Csuch that

• ui → 1 uniformly on compact subsets of G,• ui vanishes at infinity,• ui is a Herz-Schur multiplier, and supi ‖ui‖cb <∞.


ΛWH(G) ∈ [1,∞] and




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Which groups have the weak Haagerupproperty?

• Weakly amenable groups and groups with the Haagerupproperty

• Even more groups...

Theorem• If G and H are weakly Haagerup, then so is G ×H .• Given a short exact sequence

1 −→ N −→ G −→ G/N −→ 1,

if N is weakly Haagerup and G/N is amenable, then Gis weakly Haagerup.

• If Γ is a lattice in G, then Γ is weakly Haagerup if andonly if G is weakly Haagerup.



Which groups do not have the weak Haagerupproperty?

Theorem (Haagerup)The group R2 o SL(2,R) is not weakly amenable.

Theorem (with Haagerup)The group R2 o SL(2,R) does not have the weak Haagerupproperty.

Corollary (with Haagerup)The groups Z2 o SL(2,Z) and SL(n,Z) with n ≥ 3 do nothave the weak Haagerup property.



Which groups do not have the weak Haagerupproperty?

Theorem (many people)A connected simple Lie group is weakly amenable if and onlyif it has real rank zero or one.

Theorem (with Haagerup)The three groups SL(3,R), Sp(2,R) and the universalcovering group S̃p(2,R) of Sp(2,R) do not have the weakHaagerup property.

Theorem (with Haagerup)A connected simple Lie group has the weak Haagerupproperty if and only if it has real rank zero or one.



Von Neumann algebras

Let M be a von Neumann algebra with a faithful normaltracial state τ .

DefinitionM has the weak Haagerup property if there is a net (Ti)i∈Iof normal operators Ti : M → M such that

• Tix → x ultraweakly for every x ∈ M .• Ti is completely bounded, and supi ‖Ti‖cb <∞.• 〈Tix, y〉τ = 〈x,Tiy〉τ for every x, y ∈ M .• Ti extends to a compact operator on L2(M , τ),

The best (lowest) value of supi ‖Ti‖cb is the weak Haagerupconstant of M , denoted ΛWH(M ).Remark: the definition does not depend on the choice of τ .



Von Neumann algebrasLet M be a von Neumann algebra with a faithful normaltracial state τ .









The best (lowest) value of supi ‖Ti‖cb is the weak Haagerupconstant of M , denoted ΛWH(M ).

Remark: the definition does not depend on the choice of τ .



The weak Haagerup property for von Neumannalgebras

TheoremFor a discrete group Γ the following are equivalent.

• Γ has the weak Haagerup property.• L(Γ) has the weak Haagerup property.



The weak Haagerup property for von Neumannalgebras

TheoremFor a discrete group Γ the following are equivalent.

• Γ has the weak Haagerup property.• L(Γ) has the weak Haagerup property.

More precisely,

ΛWH(L(Γ)) = ΛWH(Γ).



An exampleLet Γ0 be the quaternion integer lattice in Sp(1,n) moduloits center ±I .

Set

Γ1 = Γ0 × (Z/2 o F2),Γ2 = Z2 o SL(2,Z).

Then Γ1 and Γ2 are ICC groups, and

ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,

ΛWH(Γ1) ≤ ΛWH(Γ0)ΛWH(Z/2oF2) = ΛWH(Sp(1,n))·1 ≤ 2n−1,

Remark: Γ1 and Γ2 are not weakly amenable and do nothave the Haagerup property. Both groups have AP.



An exampleLet Γ0 be the quaternion integer lattice in Sp(1,n) moduloits center ±I . Set

Γ1 = Γ0 × (Z/2 o F2),

Γ2 = Z2 o SL(2,Z).


ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,






Γ1 = Γ0 × (Z/2 o F2),Γ2 = Z2 o SL(2,Z).


ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,






Γ1 = Γ0 × (Z/2 o F2),Γ2 = Z2 o SL(2,Z).

Then Γ1 and Γ2 are ICC groups,

and

ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,






Γ1 = Γ0 × (Z/2 o F2),Γ2 = Z2 o SL(2,Z).


ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,






Γ1 = Γ0 × (Z/2 o F2),Γ2 = Z2 o SL(2,Z).


ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,


Remark: Γ1 and Γ2 are not weakly amenable and do nothave the Haagerup property.

Both groups have AP.




Γ1 = Γ0 × (Z/2 o F2),Γ2 = Z2 o SL(2,Z).


ΛWH(Γ1) ≤ 2n − 1.ΛWH(Γ2) = ∞,




university of copenhagenoperator_2014/slides/3_1... · 2014-08-13 ·...

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