Survey on Weak Amenability - Kyoto U

19 Pages·2012·0.18 MB·English

Survey on Weak Amenability
OZAWA, Narutaka
RIMSatKyotoUniversity
Conference on von Neumann Algebras and Related Topics
January 2012
ResearchpartiallysupportedbyJSPS
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 1/19
Fej´er’s theorem
T = {z ∈ C : |z| = 1}.
∞
(cid:88)
Let f ∈ C(T) and f ∼ c zk be the Fourier expansion.
k
k=−∞
m
(cid:88)
Does the partial sum s := c zk converges to f?
m k
k=−m
Not necessarily! But,
Theorem (Fej´er)
n
1 (cid:88)
The Ces´aro mean s converges to f uniformly.
m
n
m=1
n ∞
|k| 1 (cid:88) (cid:88)
Note: For ϕ (k) = (1− )∨0, one has s = ϕ (k)c zk.
n m n k
n n
m=1 k=−∞
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 2/19
(Reduced) Group C∗-algebra
Γ a countable discrete group
λ: Γ (cid:121) (cid:96) Γ the left regular representation
2
(λ ξ)(x) = ξ(s−1x),
s
(cid:88)
λ(f)ξ = f ∗ξ for f = f(s)δ ∈ CΓ
s
C∗Γ the C∗-algebra generated by λ(CΓ) ⊂ B((cid:96) Γ)
λ 2
For Γ = Z, Fourier transform (cid:96) Z ∼= L2T implements
2
C∗Z ∼= C(T), λ(f) ↔ (cid:80) f(k)zk.
λ k∈Z
Fej´er’s theorem means that multipliers
m : λ(f) (cid:55)→ λ(ϕ f)
ϕn n
converge to the identity on the group C∗-algebra C∗Z,
λ
|k|
where ϕ (k) = (1− )∨0 has finite support and is positive definite.
n n
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 3/19
Amenability
Definition
An approximate identity on Γ is a sequence (ϕ ) of finitely supported
n
functions such that ϕ → 1. A group Γ is amenable if there is an
n
approximate identity consisting of positive definite functions.
ϕ positive definite ⇔ m is completely positive on C∗Γ
ϕ λ
⇒ m is completely bounded and (cid:107)m (cid:107) = ϕ(1).
ϕ ϕ cb
A group Γ is weakly amenable (or has the Cowling–Haagerup property)
if there is an approximate identity (ϕ ) such that sup(cid:107)m (cid:107) < ∞.
n ϕn cb
Definition for locally compact groups is similar.
Fej´er’s theorem implies that Z is amenable. In fact, all abelian groups and
finite groups are amenable. The class of amenable groups is closed under
subgroups, quotients, extensions, and limits.
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 4/19
Cowling–Haagerup constant
Recall that a group Γ is weakly amenable if there is an approximate
identity (ϕ ) such that C := sup(cid:107)m (cid:107) < ∞. The optimal constant
n ϕn cb
C ≥ 1 is called the Cowling–Haagerup constant and denoted by Λ (Γ).
cb
Λ (Γ) is equal to the CBAP constant for C∗Γ and the W∗CBAP
cb λ
constant for the group von Neumann algebra LΓ:
Λ (Γ) = Λ (C∗Γ) = Λ (LΓ).
cb cb λ cb
If Λ ≤ Γ, then Λ (Λ) ≤ Λ (Γ).
cb cb
If Γ is amenable, then Λ (Γ) = 1.
cb
F is not amenable, but is weakly amenable and Λ (F ) = 1.
2 cb 2
Λ (Γ ×Γ ) = Λ (Γ )Λ (Γ ) (Cowling–Haagerup 1989).
cb 1 2 cb 1 cb 2
If Λ (Γ ) = 1, then Λ (Γ ∗Γ ) = 1 (Ricard–Xu 2006).
cb i cb 1 2
Λ is invariant under measure equivalence.
cb
OPEN: Is weak amenability preserved under free products?
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 5/19
Herz–Schur multipliers
Theorem (Grothendieck, Haagerup, Boz˙ejko–Fendler)
For a function ϕ on Γ and C ≥ 0, the following are equivalent.
The multiplier
m : λ(f) (cid:55)→ λ(ϕf)
ϕ
is completely bounded on C∗Γ and (cid:107)m (cid:107) ≤ C.
λ ϕ cb
The Schur multiplier
M : [A ] (cid:55)→ [ϕ(y−1x)A ]
ϕ x,y x,y∈Γ x,y x,y∈Γ
is bounded on B((cid:96) Γ) and (cid:107)M (cid:107) ≤ C.
2 ϕ
There are a Hilbert space H and ξ,η ∈ (cid:96) (Γ,H) such that
∞
ϕ(y−1x) = (cid:104)ξ(x),η(y)(cid:105)
and (cid:107)ξ(cid:107) (cid:107)η(cid:107) ≤ C.
∞ ∞
A function ϕ satisfying the above conditions is called a Herz–Schur
multiplier, and the optimal constant C ≥ 0 is denoted by (cid:107)ϕ(cid:107) .
cb
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 6/19
An application to convolution operators on (cid:96) Γ
p
If ϕ is a Herz–Schur multiplier, then the Schur multiplier
M : [A ] (cid:55)→ [ϕ(y−1x)A ]
ϕ x,y x,y∈Γ x,y x,y∈Γ
is bounded on B((cid:96) Γ) for all p ∈ [1,∞]. Indeed, suppose
p
ϕ(y−1x) = (cid:104)ξ(x),η(y)(cid:105) for ξ,η ∈ (cid:96) (Γ,H).
∞
Then, using H (cid:44)→ L and H (cid:44)→ L , we define V : (cid:96) Γ → (cid:96) Γ⊗L by
p q ξ p p p
V δ = δ ⊗ξ(x−1), and likewise V . One has V∗(cid:0)A⊗1(cid:1)V = M (A).
ξ x x η η ξ ϕ
Now we wonder
B((cid:96) Γ)∩ρ(Γ)(cid:48) = {λ(f) : λ(f) is bounded on (cid:96) Γ}
p p
=? SOT-cl{λ(f) : f is finitely supported}.
No counterexample is known.
Theorem (von Neumann, Cowling)
It is true if p = 2 or Γ is weakly amenable (or has the weaker property AP).
Indeed, if Γ is weakly amenable, then m (λ(f)) → λ(f) in SOT.
ϕn
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 7/19
Examples of Herz–Schur multipliers
Every coefficient of a uniformly bounded representation (π,H) of Γ is a
Herz–Schur multiplier: For every ξ,η ∈ H, the function
ϕ(s) = (cid:104)π(s)ξ,η(cid:105)
on Γ has (cid:107)ϕ(cid:107) ≤ sup(cid:107)π(x)(cid:107)2(cid:107)ξ(cid:107)(cid:107)η(cid:107). Indeed, one has
cb
ϕ(y−1x) = (cid:104)π(x)ξ,π(y−1)∗η(cid:105).
Conversely, every Herz–Schur multiplier on an amenable group Γ is a
coefficient of some unitary representation. Indeed, if Γ is amenable,
then the unit character τ is continuous on C∗Γ and
0 λ
(cid:88)
(cid:107)ω : C∗Γ (cid:51) λ(f) (cid:55)→ ϕ(s)f(s) ∈ C(cid:107) = (cid:107)τ ◦m (cid:107) ≤ (cid:107)m (cid:107).
ϕ λ 0 ϕ ϕ
For the GNS rep’n π: C∗Γ → B(H), there are ξ,η ∈ H such that
λ
(cid:104)π(λ(s))ξ,η(cid:105) = ω (λ(s)) = ϕ(s).
ϕ
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 8/19
Relation to Dixmier’s and Kadison’s problems
Thus, for any group Γ one has
B(Γ) ⊂ UB(Γ) ⊂ B (Γ),
2
where
B(Γ) the space of coefficients of unitary representations
UB(Γ) the space of coefficients of uniformly bounded representations
B (Γ) the space of Herz–Schur multipliers
2
Theorem
(Boz˙ejko 1985) A group Γ is amenable iff B(Γ) = B (Γ).
2
B(Γ) (cid:40) UB(Γ) if F (cid:44)→ Γ. ¿Is this true for any non-amenable Γ?
2
(Haagerup 1985) A Herz–Schur multiplier need not be a coefficient of
a uniformly bounded representation, i.e., UB(F ) (cid:40) B (F ).
2 2 2
A uniformly bounded representation π of Γ extends on the full group
C∗-algebra C∗Γ if all of its coefficients belong to B(Γ).
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 9/19
Examples of weakly amenable subgroups 1
Theorem (De Canni`ere–Haagerup, Cowling, Co.–Ha., Ha. 80s)
For a simple connected Lie group G, one has

1 if G = SO(1,n) or SU(1,n), Haagerup

(cid:27)
Λ (G) = 2n−1 if G = Sp(1,n), .
cb property (T)
 +∞ if rk (G) ≥ 2, e.g., SL(3,R)
R
For a lattice Γ ≤ G, one has Λ (Γ) = Λ (G).
cb cb
The idea of the proof: If G = PK with P amenable and K compact, then
for any bi-K-invariant function ϕ on G, one has
(cid:90)
(cid:107)ϕ(cid:107) = (cid:107)ϕ| (cid:107) = (cid:107)C∗(P) (cid:51) λ(f) (cid:55)→ fϕdµ ∈ C(cid:107) .
cb P cb C∗(P)∗
Further results: If rk (G) ≥ 2, then G and its lattices even fail the AP.
R
(Lafforgue–de la Salle 2010 and Haagerup–de Laat 2012).
N.OZAWA (RIMSatKyotoUniversity) WeakAmenability 10/19