Amenability of operator algebras on Banach spaces, I

22 Pages·2010·0.15 MB·English

by  Volker Runde

Amenabilityof
operator
algebrason
Banach
spaces,I
VolkerRunde Amenability of operator algebras on Banach
Prelude spaces, I
Amenable
groups
Amenable
Banach
algebras Volker Runde
C∗-andvon
Neumann
UniversityofAlberta
algebras
Similarity
problems NBFAS, Leeds, June 1, 2010
Philosophical musings
Amenabilityof
operator
algebrason
Banach
spaces,I A quote
VolkerRunde
Big things are fucking rarely amenable!
Prelude
Amenable
groups N.N., Istanbul, 2004.
Amenable
Banach
algebras
C∗-andvon Thus...
Neumann
algebras
AMENABLE ≈ SMALL
Similarity
problems
Mean things
Amenabilityof
operator
algebrason Definition
Banach
spaces,I
Let G be a locally compact group. A mean on L∞(G) is a
VolkerRunde
functional M ∈ L∞(G)∗ such that (cid:104)1,M(cid:105) = (cid:107)M(cid:107) = 1.
Prelude
Amenable
Definition (J. von Neumann 1929; M. M. Day, 1949)
groups
Amenable G is amenable if there is a mean on L∞(G) which is left
Banach
algebras invariant, i.e.,
C∗-andvon
Neumann
algebras (cid:104)L φ,M(cid:105) = (cid:104)φ,M(cid:105) (x ∈ G, φ ∈ L∞(G)),
x
Similarity
problems
where
(L φ)(y) := φ(xy) (y ∈ G).
x
Some amenable groups...
Amenabilityof
Examples
operator
algebrason
Banach
spaces,I 1 Compact groups amenable: M = Haar measure.
VolkerRunde
2 Abelian groups are amenable: use Markov–Kakutani to get
Prelude M.
Amenable
groups
Amenable Really nice hereditary properties!
Banach
algebras
C∗-andvon 1 If G is amenable and H < G, then H is amenable.
Neumann
algebras 2 If G is is amenable and N (cid:67)G, then G/N is amenable.
Sprimobillaermitsy 3 If G and N (cid:67)G are such that N and G/N are amenable,
then G is amenable.
4 If (Hα)α is a directed union of closed, amenable subgroups
(cid:83)
of G such that G = H , then G is amenable.
α α
...and a non-amenable one, I
Amenabilityof
operator
algebrason
Banach
spaces,I Example
VolkerRunde
Let F be the free group in two generators.
2
Prelude Assume that there is a left invariant mean M on (cid:96)∞(F ).
2
Amenable Define
groups
µ : P(F ) → [0,1], E (cid:55)→ (cid:104)χ ,M(cid:105).
Amenable 2 E
Banach
algebras Then
C∗-andvon
Neumann µ is finitely additive,
algebras
µ(F ) = 1, and
Similarity 2
problems
µ(xE) = µ(E) (x ∈ F , E ⊂ F ).
2 2
...and a non-amenable one, II
Amenabilityof
operator Example (continued...)
algebrason
Banach
spaces,I For x ∈ {a,b,a−1,b−1} set
VolkerRunde
W(x) := {w ∈ F : w starts with x}.
Prelude 2
Amenable
groups Let w ∈ F2\W(a). Then a−1w ∈ W(a−1), therefore
Amenable
Banach w ∈ aW(a−1),
algebras
C∗-andvon
Neumann and thus
algebras
F = W(a)∪aW(a−1).
Similarity 2
problems
Similarly,
F = W(b)∪bW(b−1)
2
holds.
...and a non-amenable one, III
Amenabilityof
operator Example (continued even further)
algebrason
Banach
spaces,I Since
VolkerRunde
· · · ·
F = {(cid:15)} ∪ W(a) ∪ W(a−1) ∪ W(b) ∪ W(b−1),
Prelude 2
Amenable
groups we have
Amenable
Banach 1 = µ(F )
algebras 2
C∗-andvon ≥ µ(W(a))+µ(aW(a−1))+µ(W(b))+µ(bW(b−1))
Neumann
algebras ≥ µ(W(a)∪aW(a−1))+µ(W(b)∪bW(b−1))
Similarity
problems = µ(F )+µ(F )
2 2
= 2,
which is nonsense.
Consequences
Amenabilityof
operator
algebrason
Banach Hence, G is amenable if...
spaces,I
VolkerRunde 1 G is solvable or
Prelude 2 G is locally finite.
Amenable
groups
Amenable But G is not amenable if...
Banach
algebras G contains F as closed subgroup, e.g., if
2
C∗-andvon
Neumann G = SL(N,R) with N ≥ 2,
algebras
Similarity G = GL(N,R) with N ≥ 2, or
problems
G = SO(N,R) with N ≥ 3 equipped with the discrete
topology.
Amenable Banach algebras via approximate
diagonals
Amenabilityof
operator
algebrason
Banach
spaces,I Definition (B. E. Johnson, 1972)
VolkerRunde
A Banach algebra A is said to be amenable if it possesses an
Prelude
approximate diagonal, i.e., a bounded net (d ) in the
α α
Amenable
groups projective tensor product A⊗ˆA such that
Amenable
Banach a·d −d ·a → 0 (a ∈ A)
algebras α α
C∗-andvon
Neumann and
algebras
a∆d → a (a ∈ A)
Similarity α
problems
with ∆ : A⊗ˆA → A denoting multiplication.
Amenable Banach algebras and amenable groups
Amenabilityof
operator
algebrason
Banach
spaces,I
Theorem (B. E. Johnson, 1972)
VolkerRunde
Prelude The following are equivalent for a locally compact group G:
Amenable 1 G is amenable;
groups
Amenable 2 L1(G) is amenable.
Banach
algebras
C∗-andvon Grand theme
Neumann
algebras
Let C be a class of Banach algebras. Characterize the amenable
Similarity
problems members of C!