Amenability of operator algebras on Banach spaces, II

30 Pages·2010·0.2 MB·English

by  Volker Runde

Amenabilityof
operator
algebrason
Banach
spaces,II
VolkerRunde Amenability of operator algebras on Banach
Amenabilityof spaces, II
K(E)
Amenabilityof
B(E)
Apositive
example
B((cid:96)p⊕(cid:96)q) Volker Runde
withp(cid:54)=p
B((cid:96)p)
UniversityofAlberta
NBFAS, Leeds, June 1, 2010
The finite-dimensional case
Amenabilityof
operator Example
algebrason
Banach Let E be a Banach space with n := dimE < ∞ so that
spaces,II
VolkerRunde
∼
B(E) = K(E) = M .
n
Amenabilityof
K(E)
Let G be a finite subgroup of invertible elements of M such
Amenabilityof n
B(E) that spanG = M .
n
Apositive
example Set
BBwi((t(cid:96)(cid:96)hppp)⊕(cid:54)=(cid:96)pq) d := 1 (cid:88)g ⊗g−1.
|G|
g∈G
Then
a·d = d·a (a ∈ M )
n
and ∆d = I .
n
Hence, K(E) = B(E) is amenable.
Some more results
Amenabilityof
Theorem (B. E. Johnson, 1972)
operator
algebrason
Banach K(E) is amenable if E = (cid:96)p with 1 < p < ∞ or E = C(T).
spaces,II
VolkerRunde
Amenable Banach algebras must have bounded approximate
Amenabilityof identities...
K(E)
Amenabilityof Theorem (N. Grønbæk & G. A. Willis, 1994)
B(E)
Apositive
example Suppose that E has the approximation property. Then K(E)
B((cid:96)p⊕(cid:96)q)
withp(cid:54)=p has a bounded approximate identity if and only if E∗ has the
B((cid:96)p)
bounded approximation property.
Example
Let E = (cid:96)2⊗ˆ(cid:96)2. Then E has the approximation property, but
E∗ = B((cid:96)2) doesn’t. Hence, K(E) does not have a bounded
approximate identity and is thus not amenable.
Finite, biorthogonal systems
Amenabilityof
operator
algebrason Definition
Banach
spaces,II A finite, biorthogonal system is a set
VolkerRunde {(x ,φ ) : j,k = 1,...,n} ⊂ E ×E∗ such that
j k
Amenabilityof
K(E) (cid:104)x ,φ (cid:105) = δ (j,k = 1,...,n).
j k j,k
Amenabilityof
B(E)
Apositive
example
B((cid:96)p⊕(cid:96)q) Remark
withp(cid:54)=p
B((cid:96)p)
If {(x ,φ ) : j,k = 1,...,n} is a finite, biorthogonal system,
j k
then
n
(cid:88)
θ : M → F(E), [α ] (cid:55)→ α x ⊗φ
n j,k j,k j k
j,k=1
is an algebra homomorphism.
Property (A)
Amenabilityof
operator
algebrason Definition (N. Grønbæk, BEJ, & G. A. Willis, 1994)
Banach
spaces,II
We say that E has property (A) if there is a net
VolkerRunde
({(x ,φ ) : j,k = 1,...,n }) of finite biorthogonal systems
j,λ k,λ λ λ
Amenabilityof with corresponding homomorphisms θ : M → F(E) with the
K(E) λ nλ
following properties:
Amenabilityof
B(E)
Aexapmospitleive 1 θλ(Inλ) → idE uniformly on compacts;
BBwi((t(cid:96)(cid:96)hppp)⊕(cid:54)=(cid:96)pq) 2 θλ(Inλ)∗ → idE∗ uniformly on compacts;
3 for each λ, there is a finite group Gλ of invertible elements
of M spanning M such that
n n
λ λ
supmax(cid:107)θ (g)(cid:107) < ∞.
λ
λ g∈Gλ
Property (A) and the amenability of K(E)
Amenabilityof
operator
algebrason The idea behind (A)
Banach
spaces,II
Use the diagonals of the M ’s to construct an approximate
n
VolkerRunde λ
diagonal for K(E).
Amenabilityof
K(E)
Theorem (N. Grønbæk, BEJ, & G. A. Willis, 1994)
Amenabilityof
B(E)
Apositive Suppose that E has property (A). Then K(E) is amenable.
example
B((cid:96)p⊕(cid:96)q)
withp(cid:54)=p
B((cid:96)p)
Examples
1 Lp(µ) has property (A) for all 1 ≤ p < ∞ and all µ.
2 C(K) has property (A) for each compact K, as does
therefore L∞(µ) for each µ.
The “scalar plus compact” problem
Amenabilityof Question
operator
algebrason
Banach Is there an infinite-dimensional Banach space E such that
spaces,II
B(E) = K(E)+Cid ?
VolkerRunde E
Amenabilityof Theorem (S. A. Argyros & R. G. Haydon, 2009)
K(E)
Amenabilityof There is a Banach space E such that B(E) = K(E)+Cid
B(E) E
Apositive and E∗ = (cid:96)1.
example
B((cid:96)p⊕(cid:96)q)
withp(cid:54)=p
B((cid:96)p) Theorem (N. Grønbæk, BEJ, & G. A. Willis, 1994)
Suppose that E∗ has property (A). Then so has E.
Corollary
There is an infinite-dimensional Banach space E such that
B(E) is amenable.
Non-amenability of B((cid:96)p ⊕ (cid:96)q) for p (cid:54)= q, I
Amenabilityof
operator Theorem (G. A. Willis, unpublished)
algebrason
Banach
spaces,II Let p,q ∈ (1,∞) be such that p (cid:54)= q. Then B((cid:96)p ⊕(cid:96)q) is not
VolkerRunde amenable.
Amenabilityof
K(E) Ingredients
Amenabilityof
B(E)
Apositive 1 A quotient of an amenable Banach algebra is again
example
B((cid:96)p⊕(cid:96)q) amenable.
withp(cid:54)=p
B((cid:96)p)
2 Every complemented closed ideal of an amenable Banach
algebra is amenable.
3 Every amenable Banach algebra has a bounded
approximate identity.
4 Pitt’s Theorem. If p > q, then B((cid:96)p,(cid:96)q) = K((cid:96)p,(cid:96)q).
Non-amenability of B((cid:96)p ⊕ (cid:96)q) for p (cid:54)= q, II
Amenabilityof
Proof.
operator
algebrason
Banach Suppose that p > q. Note that
spaces,II
VolkerRunde (cid:20) B((cid:96)p) B((cid:96)q,(cid:96)p)(cid:21)
B((cid:96)p ⊕(cid:96)q) =
Amenabilityof B((cid:96)p,(cid:96)q)K((cid:96)p,(cid:96)q) B((cid:96)q)
K(E)
Amenabilityof and
B(E) (cid:20) K((cid:96)p) K((cid:96)q,(cid:96)p)(cid:21)
Aexapmospitleive K((cid:96)p ⊕(cid:96)q) = ,
B((cid:96)p⊕(cid:96)q) K((cid:96)p,(cid:96)q) K((cid:96)q)
withp(cid:54)=p
B((cid:96)p)
so that
(cid:20)C((cid:96)p) ∗ (cid:21)
C((cid:96)p ⊕(cid:96)q) = .
0 C((cid:96)q)
(cid:20) (cid:21)
0 ∗
Then I := (cid:54)= {0} is a complemented ideal of C((cid:96)p⊕(cid:96)q),
0 0
thus is amenable, and thus has a BAI. But I2 = {0}...
Non-amenability of B((cid:96)p) for p = 1,2,∞
Amenabilityof
operator
algebrason
Banach Theorem (C. J. Read, <2006)
spaces,II
VolkerRunde B((cid:96)1) is not amenable.
Amenabilityof
K(E) Progress since
Amenabilityof
B(E)
Apositive 1 Simplification of Read’s proof by G. Pisier, 2004.
example
Bwi(t(cid:96)hpp⊕(cid:54)=(cid:96)pq) 2 Simultaneous proof for the non-amenability of B((cid:96)p) for
B((cid:96)p)
p = 1,2,∞ by N. Ozawa, 2006.
Question
Is B((cid:96)p) amenable for any p ∈ (1,∞)\{2}?