Amenability of operator algebras on Banach spaces, I · Amenability of operator algebras on Banach...

transcript

Amenability ofoperator

algebras onBanachspaces, I

Volker Runde

Prelude

Amenablegroups

AmenableBanachalgebras

C∗- and vonNeumannalgebras

Similarityproblems

Amenability of operator algebras on Banachspaces, I

Volker Runde

University of Alberta

NBFAS, Leeds, June 1, 2010

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Philosophical musings

A quote

Big things are rarely amenable!

N.N., Istanbul, 2004.

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

A quote

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

A quote

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

A quote

Istanbul, 2004.

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

A quote

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

A quote

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

A quote

Thus. . .

AMENABLE

≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

A quote

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

A quote

Big things are fucking rarely amenable!

Thus. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Mean things

Definition

Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.

Definition (J. von Neumann 1929; M. M. Day, 1949)

G is amenable if there is a mean on L∞(G ) which is leftinvariant, i.e.,

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

where(Lxφ)(y) := φ(xy) (y ∈ G ).

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

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Amenablegroups

Similarityproblems

Mean things

Definition

Let G be a locally compact group.

A mean on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Mean things

Definition

Let G be a locally compact group. A mean

on L∞(G ) is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Mean things

Definition

Let G be a locally compact group. A mean on L∞(G )

is afunctional M ∈ L∞(G )∗ such that 〈1,M〉 = ‖M‖ = 1.

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Mean things

Definition

Let G be a locally compact group. A mean on L∞(G ) is afunctional M ∈ L∞(G )∗

such that 〈1,M〉 = ‖M‖ = 1.

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

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Amenablegroups

Similarityproblems

Mean things

Definition

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

G is amenable

if there is a mean on L∞(G ) which is leftinvariant, i.e.,

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

G is amenable if there is a mean on L∞(G )

which is leftinvariant, i.e.,

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

G is amenable if there is a mean on L∞(G ) which is leftinvariant,

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

Similarityproblems

Mean things

Definition

〈Lxφ,M〉 = 〈φ,M〉 (x ∈ G , φ ∈ L∞(G )),

Volker Runde

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Amenablegroups

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Some amenable groups. . .

Examples

1 Compact groups amenable: M = Haar measure.

2 Abelian groups are amenable: use Markov–Kakutani to getM.

Really nice hereditary properties!

1 If G is amenable and H < G , then H is amenable.

2 If G is is amenable and N C G , then G/N is amenable.

3 If G and N C 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 subgroupsof G such that G =

⋃α Hα, then G is amenable.

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Amenablegroups

Similarityproblems

Examples

Volker Runde

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Amenablegroups

Similarityproblems

Examples

1 Compact groups amenable:

M = Haar measure.

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Amenablegroups

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Examples

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Amenablegroups

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Examples

2 Abelian groups are amenable:

use Markov–Kakutani to getM.

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Amenablegroups

Similarityproblems

Examples

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Amenablegroups

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Examples

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Examples

1 If G is amenable

and H < G , then H is amenable.

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Amenablegroups

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Examples

1 If G is amenable and H < G ,

then H is amenable.

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Examples

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Examples

2 If G is is amenable

and N C G , then G/N is amenable.

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Examples

2 If G is is amenable and N C G ,

then G/N is amenable.

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Examples

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Examples

3 If G

and N C G are such that N and G/N are amenable,then G is amenable.

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Examples

3 If G and N C G

are such that N and G/N are amenable,then G is amenable.

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Examples

3 If G and N C G are such that N and G/N are amenable,

then G is amenable.

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Amenablegroups

Similarityproblems

Examples

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Amenablegroups

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Examples

4 If (Hα)α is a directed union

of closed, amenable subgroupsof G such that G =

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Amenablegroups

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Examples

4 If (Hα)α is a directed union of closed, amenable subgroupsof G

such that G =⋃α Hα, then G is amenable.

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Examples

⋃α Hα,

then G is amenable.

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Examples

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. . . and a non-amenable one, I

Example

Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).Define

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ is finitely additive,

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

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Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

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Amenablegroups

Similarityproblems

Example

Let F2 be the free group in two generators.

Assume that there is a left invariant mean M on `∞(F2).Define

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Example

Let F2 be the free group in two generators.Assume that there is a left invariant mean M on `∞(F2).

Defineµ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

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Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

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Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

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Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1,

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Example

µ : P(F2)→ [0, 1], E 7→ 〈χE ,M〉.

µ(F2) = 1, and

µ(xE ) = µ(E ) (x ∈ F2, E ⊂ F2).

Volker Runde

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Amenablegroups

Similarityproblems

. . . and a non-amenable one, II

Example (continued. . . )

For x ∈ {a, b, a−1, b−1} set

W (x) := {w ∈ F2 : w starts with x}.

Let w ∈ F2 \W (a). Then a−1w ∈W (a−1), therefore

w ∈ aW (a−1),

and thusF2 = W (a) ∪ aW (a−1).

Similarly,F2 = W (b) ∪ bW (b−1)

holds.

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

w ∈ aW (a−1),

holds.

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Similarityproblems

For x ∈ {a, b, a−1, b−1}

w ∈ aW (a−1),

holds.

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

w ∈ aW (a−1),

holds.

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

Let w ∈ F2 \W (a).

Then a−1w ∈W (a−1), therefore

w ∈ aW (a−1),

holds.

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

Let w ∈ F2 \W (a). Then a−1w ∈W (a−1),

therefore

w ∈ aW (a−1),

holds.

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

w ∈ aW (a−1),

holds.

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

w ∈ aW (a−1),

holds.

Volker Runde

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Amenablegroups

Similarityproblems

For x ∈ {a, b, a−1, b−1} set

w ∈ aW (a−1),

holds.

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Amenablegroups

Similarityproblems

. . . and a non-amenable one, III

Example (continued even further)

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

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Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(

W (a−1)) + µ(W (b)) + µ(

W (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

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Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

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Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

Volker Runde

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Amenablegroups

Similarityproblems

F2 = {ε}·∪W (a)

·∪W (a−1)

·∪W (b)

·∪W (b−1),

we have

1 = µ(F2)

≥ µ(W (a)) + µ(aW (a−1)) + µ(W (b)) + µ(bW (b−1))

≥ µ(W (a) ∪ aW (a−1)) + µ(W (b) ∪ bW (b−1))

= µ(F2) + µ(F2)

which is nonsense.

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Amenablegroups

Similarityproblems

Consequences

Hence, G is amenable if. . .

1 G is solvable or

2 G is locally finite.

But G is not amenable if. . .

G contains F2 as closed subgroup, e.g., if

G = SL(N,R) with N ≥ 2,

G = GL(N,R) with N ≥ 2, or

G = SO(N,R) with N ≥ 3 equipped with the discretetopology.

Volker Runde

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Amenablegroups

Similarityproblems

Consequences

1 G is solvable or

Volker Runde

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Amenablegroups

Similarityproblems

Consequences

1 G is solvable

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Amenablegroups

Similarityproblems

Consequences

1 G is solvable or

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Consequences

1 G is solvable or

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Amenablegroups

Similarityproblems

Consequences

1 G is solvable or

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Consequences

1 G is solvable or

G contains F2 as closed subgroup,

e.g., if

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Consequences

1 G is solvable or

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Consequences

1 G is solvable or

G = SL(N,R)

with N ≥ 2,

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Similarityproblems

Consequences

1 G is solvable or

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Consequences

1 G is solvable or

G = GL(N,R) with N ≥ 2,

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Similarityproblems

Consequences

1 G is solvable or

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Similarityproblems

Consequences

1 G is solvable or

G = SO(N,R)

with N ≥ 3 equipped with the discretetopology.

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Amenablegroups

Similarityproblems

Consequences

1 G is solvable or

G = SO(N,R) with N ≥ 3

equipped with the discretetopology.

Volker Runde

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Amenablegroups

Similarityproblems

Consequences

1 G is solvable or

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Amenablegroups

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Amenable Banach algebras via approximatediagonals

Definition (B. E. Johnson, 1972)

A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that

a · dα − dα · a→ 0 (a ∈ A)

anda∆dα → a (a ∈ A)

with ∆ : A⊗A→ A denoting multiplication.

Volker Runde

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Amenablegroups

Similarityproblems

a · dα − dα · a→ 0 (a ∈ A)

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Amenablegroups

Similarityproblems

A Banach algebra A is said to be amenable

if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that

a · dα − dα · a→ 0 (a ∈ A)

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Amenablegroups

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A Banach algebra A is said to be amenable if it possesses anapproximate diagonal,

i.e., a bounded net (dα)α in theprojective tensor product A⊗A such that

a · dα − dα · a→ 0 (a ∈ A)

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Amenablegroups

Similarityproblems

A Banach algebra A is said to be amenable if it possesses anapproximate diagonal, i.e., a bounded net (dα)α in theprojective tensor product A⊗A

such that

a · dα − dα · a→ 0 (a ∈ A)

Volker Runde

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Amenablegroups

Similarityproblems

a · dα − dα · a→ 0 (a ∈ A)

Volker Runde

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Amenablegroups

Similarityproblems

a · dα − dα · a→ 0 (a ∈ A)

Volker Runde

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Amenablegroups

Similarityproblems

a · dα − dα · a→ 0 (a ∈ A)

Volker Runde

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Amenablegroups

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Amenable Banach algebras and amenable groups

Theorem (B. E. Johnson, 1972)

The following are equivalent for a locally compact group G :

1 G is amenable;

2 L1(G ) is amenable.

Grand theme

Let C be a class of Banach algebras. Characterize the amenablemembers of C!

Volker Runde

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

Volker Runde

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

Volker Runde

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

Volker Runde

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

Let C be a class of Banach algebras.

Characterize the amenablemembers of C!

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Amenablegroups

Similarityproblems

1 G is amenable;

Grand theme

Volker Runde

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Amenablegroups

Similarityproblems

More from abstract harmonic analysis

Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemskiı,2002)

The following are equivalent:

1 M(G ) is amenable;

2 G is amenable and discrete.

Theorem (B. E. Forrest & VR, 2005)

1 A(G ) is amenable;

2 G has an abelian subgroup of finite index.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

2 G is amenable

and discrete.

Volker Runde

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Amenablegroups

Similarityproblems

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

2 G has an abelian subgroup

of finite index.

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Amenablegroups

Similarityproblems

Volker Runde

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Amenablegroups

Similarityproblems

From old to new. . .

Hereditary properties

1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.

2 If I C A is such that both I and A/I are amenable, then A

is amenable.

3 If A is amenable and I C A, then the following areequivalent:

1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented

in A∗.

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Amenablegroups

Similarityproblems

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

1 If A is amenable

and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A, then A/I is amenable.

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range,

then B is amenable. Inparticular, if I C A, then A/I is amenable.

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable.

Inparticular, if I C A, then A/I is amenable.

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular,

if I C A, then A/I is amenable.

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 If A is amenable and θ : A→ B is a boundedhomomorphism with dense range, then B is amenable. Inparticular, if I C A,

then A/I is amenable.

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

2 If I C A is such that

both I and A/I are amenable, then A

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

2 If I C A is such that both I

and A/I are amenable, then A

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

2 If I C A is such that both I and A/I

are amenable, then A

is amenable.

in A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

2 If I C A is such that both I and A/I are amenable,

then A

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

3 If A is amenable

and I C A, then the following areequivalent:

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

3 If A is amenable and I C A,

then the following areequivalent:

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

1 I is amenable;

2 I has a bounded approximate identity;3 I is weakly complemented in A, i.e., I⊥ is complemented

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

is amenable.

1 I is amenable;2 I has a bounded approximate identity;

3 I is weakly complemented in A, i.e., I⊥ is complementedin A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

is amenable.

1 I is amenable;2 I has a bounded approximate identity;3 I is weakly complemented in A,

i.e., I⊥ is complementedin A∗.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

is amenable.

in A∗.

Volker Runde

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Amenablegroups

Similarityproblems

Grand theme, reprise!

The “meaning” of amenability

What does it mean for a member of a class C of Banachalgebras to be amenable for the following classes C?

1 all C ∗-algebras;

2 all von Neumann algebras;

3 all norm closed, but not necessarily self-adjointsubalgebras of B(H);

4 all algebras K(E );

5 all algebras B(E ).

Remember. . .

AMENABLE ≈ SMALL

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Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

What does it mean for a member of a class C of Banachalgebras to be amenable

for the following classes C?

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

3 all norm closed,

but not necessarily self-adjointsubalgebras of B(H);

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

3 all norm closed, but not necessarily self-adjoint

subalgebras of B(H);

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Remember. . .

AMENABLE ≈ SMALL

Volker Runde

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Amenablegroups

Similarityproblems

Complete positivity

If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,

T (n)[aj ,k ] := [Taj ,k ] ([aj ,k ] ∈ Mn(A)).

Definition

T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.

Example

M2 → M2, a 7→ at

is positive, but not completely positive.

Volker Runde

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Amenablegroups

Similarityproblems

Complete positivity

If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N.

Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,

Definition

Example

M2 → M2, a 7→ at

Volker Runde

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Amenablegroups

Similarityproblems

Complete positivity

If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear,

we writeT (n) : Mn(A)→ Mn(B) for its amplification, i.e.,

Definition

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

If A and B are C ∗-algebras, then so are Mn(A) and Mn(B) forall n ∈ N. Whenever T : A→ B is linear, we writeT (n) : Mn(A)→ Mn(B) for its amplification,

Definition

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

Volker Runde

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Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

Volker Runde

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Amenablegroups

Similarityproblems

Complete positivity

Definition

T : A→ B is called completely positive

ifT (n) : Mn(A)→ Mn(B) is positive for each n ∈ N.

Example

M2 → M2, a 7→ at

Volker Runde

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Amenablegroups

Similarityproblems

Complete positivity

Definition

T : A→ B is called completely positive ifT (n) : Mn(A)→ Mn(B) is positive

for each n ∈ N.

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

is positive,

but not completely positive.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Complete positivity

Definition

Example

M2 → M2, a 7→ at

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Nuclear C ∗-algebras

Definition

A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers and of completely positive contractions

such that(Sλ ◦ Tλ)a→ a (a ∈ A).

Volker Runde

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Amenablegroups

Similarityproblems

Definition

Volker Runde

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Amenablegroups

Similarityproblems

Definition

A C ∗-algebra A is called nuclear

if there are nets (nλ)λ ofpositive integers and of completely positive contractions

Volker Runde

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Amenablegroups

Similarityproblems

Definition

A C ∗-algebra A is called nuclear if there are nets (nλ)λ ofpositive integers

and of completely positive contractions

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Volker Runde

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Amenablegroups

Similarityproblems

Definition

Volker Runde

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Amenablegroups

Similarityproblems

Definition

Volker Runde

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Amenablegroups

Similarityproblems

Nuclearity and amenability

Theorem (A. Connes, 1978)

A C ∗-algebra A is nuclear if it is amenable and A∗ is separable.

Theorem (U. Haagerup, 1983)

All nuclear C ∗-algebras are amenable.

Theorem (A. Connes, U. Haagerup, et al.)

The following are equivalent for a C ∗-algebra A:

1 A is nuclear;

2 A is amenable.

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

A C ∗-algebra A is nuclear if it is amenable

and A∗ is separable.

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

1 A is nuclear;

2 A is amenable.

Volker Runde

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Amenablegroups

Similarityproblems

Nuclear von Neumann algebras

Theorem (S. Wasserman, 1976)

The following are equivalent for a von Neumann algebra M:

1 M is nuclear;

2 M is subhomogeneous, i.e.,

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

with n1, . . . , nk ∈ N and M1, . . . ,Mk abelian.

Corollary

B(H) is amenable if and only if dim H <∞.

Volker Runde

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

Volker Runde

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

Volker Runde

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

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Amenablegroups

Similarityproblems

1 M is nuclear;

2 M is subhomogeneous,

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

with n1, . . . , nk ∈ N

and M1, . . . ,Mk abelian.

Corollary

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

Volker Runde

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

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Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

B(H) is amenable

if and only if dim H <∞.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

1 M is nuclear;

M ∼= Mn1(M1)⊕ · · · ⊕Mnk(Mk)

Corollary

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Representations of locally compact groups

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

1 unitary if π(G ) consists of unitaries,

2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and

3 uniformly bounded if

supg∈G‖π(g)‖ <∞.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

Volker Runde

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Amenablegroups

Similarityproblems

Definition

A representation of G

on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H

is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π

from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G

into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H)

which is continuous with respect to the given topologyon G and the operator topology on B(H).We call π:

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G

and the operator topology on B(H).We call π:

Volker Runde

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Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the weak operator topology on B(H).

We call π:

Volker Runde

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Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).

We call π:

Volker Runde

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Amenablegroups

Similarityproblems

Definition

A representation of G on a Hilbert space H is a grouphomomomorphism π from G into the invertible elements ofB(H) which is continuous with respect to the given topologyon G and the strong operator topology on B(H).We call π:

Volker Runde

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Amenablegroups

Similarityproblems

Definition

1 unitary

if π(G ) consists of unitaries,

Volker Runde

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Amenablegroups

Similarityproblems

Definition

Volker Runde

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Amenablegroups

Similarityproblems

Definition

2 unitarizable

if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and

Volker Runde

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Amenablegroups

Similarityproblems

Definition

2 unitarizable if π is similar to a unitary representation,

i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary, and

Volker Runde

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Amenablegroups

Similarityproblems

Definition

2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H)

such that T−1π(·)T isunitary, and

Volker Runde

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Amenablegroups

Similarityproblems

Definition

2 unitarizable if π is similar to a unitary representation, i.e.,there is an invertible T ∈ B(H) such that T−1π(·)T isunitary,

Volker Runde

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Amenablegroups

Similarityproblems

Definition

3 uniformly bounded

Volker Runde

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Amenablegroups

Similarityproblems

Definition

Volker Runde

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Amenablegroups

Similarityproblems

Unitarizability and amenability

Obvious. . .

π unitary =⇒ π unitarizable =⇒ π uniformly bounded.

Theorem (J. Dixmier, 1950)

Suppose that G is amenable. Then every uniformly boundedrepresentation of G is unitarizable.

Big open question

Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?

It’s false for F2!

Volker Runde

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Amenablegroups

Similarityproblems

Obvious. . .

π unitary

=⇒ π unitarizable =⇒ π uniformly bounded.

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

π unitary =⇒ π unitarizable

=⇒ π uniformly bounded.

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Suppose that G is amenable.

Then every uniformly boundedrepresentation of G is unitarizable.

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Suppose that G is amenable. Then every uniformly boundedrepresentation of G

is unitarizable.

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Does the converse hold,

i.e., is any G such that each uniformlybounded represenation is unitarizable already amenable?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Does the converse hold, i.e., is any G

such that each uniformlybounded represenation is unitarizable already amenable?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Does the converse hold, i.e., is any G such that each uniformlybounded represenation is unitarizable

already amenable?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Obvious. . .

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

From groups to algebras

Integration of representations

If π is a uniformly bounded representation of G , we canintegrate π and obtain a representation of L1(G ) on H:

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

If G is amenable, then A := π(L1(G )) is amenable.

Slightly more difficult

If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

If π is a uniformly bounded representation of G ,

we canintegrate π and obtain a representation of L1(G ) on H:

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

If π is a uniformly bounded representation of G , we canintegrate π

and obtain a representation of L1(G ) on H:

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

If G is amenable,

then A := π(L1(G )) is amenable.

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

If G is amenable,

then π is unitarizable, so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

If G is amenable, then π is unitarizable,

so that there is aninvertible T ∈ B(H) such that TAT−1 is a C ∗-subalgebra ofB(H).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

If G is amenable, then π is unitarizable, so that there is aninvertible T ∈ B(H)

such that TAT−1 is a C ∗-subalgebra ofB(H).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

π(f ) :=

f (g)π(g) dg (f ∈ L1(G )).

Volker Runde

Prelude

Amenablegroups

Similarityproblems

The similarity problem for amenable operatoralgebras

Definition

A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).

Big open question

Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A closed,

but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A closed, but not necessarily self-adjoint

subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A closed, but not necessarily self-adjoint subalgebra of B(H)

iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra

if there is an invertible T ∈ B(H)such that TAT−1 is a C ∗-subalgebra of B(H).

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

A closed, but not necessarily self-adjoint subalgebra of B(H) iscalled similar to a C ∗-algebra if there is an invertible T ∈ B(H)

such that TAT−1 is a C ∗-subalgebra of B(H).

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Big open question

Is every closed, amenable subalgebra of B(H)

similar to aC ∗-subalgebra of B(H) (which is necessarily nuclear)?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Big open question

Is every closed, amenable subalgebra of B(H) similar to aC ∗-subalgebra of B(H)

(which is necessarily nuclear)?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Big open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Some partial results

Theorem (J. A. Gifford, <2006)

Suppose that A is a closed, amenable subalgebra of K(H).Then A is similar to a C ∗-subalgebra of K(H).

Definition

Let C ≥ 1. We call A C -amenable if A has an approximatediagonal bounded by C .

Theorem (D. Blecher & C. LeMerdy, 2004)

Let A be a closed, 1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.

Open question

What if A is commutative, even generated by one operator?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Suppose that A is a closed, amenable subalgebra

of K(H).Then A is similar to a C ∗-subalgebra of K(H).

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Suppose that A is a closed, amenable subalgebra of K(H).

Then A is similar to a C ∗-subalgebra of K(H).

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Let C ≥ 1.

We call A C -amenable if A has an approximatediagonal bounded by C .

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Let C ≥ 1. We call A C -amenable

if A has an approximatediagonal bounded by C .

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Let A be a closed,

1-amenable subalgebra of B(H). Then A isa nuclear C ∗-algebra.

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Let A be a closed, 1-amenable

subalgebra of B(H). Then A isa nuclear C ∗-algebra.

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Let A be a closed, 1-amenable subalgebra of B(H).

Then A isa nuclear C ∗-algebra.

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

What if A is commutative,

even generated by one operator?

Volker Runde

Prelude

Amenablegroups

Similarityproblems

Definition

Open question

Amenability of operator algebras on Banach spaces, I · Amenability of operator algebras on Banach...

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