Unique amenability of topological groups Dana Barto sov a - - PowerPoint PPT Presentation

Unique amenability of topological groups

Dana Bartoˇ sov´ a

Carnegie Mellon University

BLAST University of Denver August 6, 2018

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

G × X

X - a continuous action

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space g(hx) = (gh)x ex = x

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space g(hx) = (gh)x ex = x (g, ·) : X

X- a homeomorphism

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space g(hx) = (gh)x ex = x (g, ·) : X

X- a homeomorphism

G

Homeo(X) - a continuous homomorphism

Dana Bartoˇ sov´ a Unique amenability of topological groups

G-flow

G × X

X - a continuous action

↑ ↑ topological compact group Hausdorff space g(hx) = (gh)x ex = x (g, ·) : X

X- a homeomorphism

G

Homeo(X) - a continuous homomorphism

We call X a G-flow.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples

f : X

X homeomorphism, X compact Hausdorff

Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples

f : X

X homeomorphism, X compact Hausdorff

gives rise to a Z-flow Z × X

X, (n, x) → fz(x)

Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples

f : X

X homeomorphism, X compact Hausdorff

gives rise to a Z-flow Z × X

X, (n, x) → fz(x)

1 rotation of a circle ρα : R/Z

R/Z, x → x + α

Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples

f : X

X homeomorphism, X compact Hausdorff

gives rise to a Z-flow Z × X

X, (n, x) → fz(x)

1 rotation of a circle ρα : R/Z

R/Z, x → x + α

2 Bernoulli shift σ : 2Z

2Z, σ(f)(n) = f(n + 1)

Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples

f : X

X homeomorphism, X compact Hausdorff

gives rise to a Z-flow Z × X

X, (n, x) → fz(x)

1 rotation of a circle ρα : R/Z

R/Z, x → x + α

2 Bernoulli shift σ : 2Z

2Z, σ(f)(n) = f(n + 1)

3 1+ : βZ

βZ, 1 + u = {1 + S, S ∈ u}

Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples

f : X

X homeomorphism, X compact Hausdorff

gives rise to a Z-flow Z × X

X, (n, x) → fz(x)

1 rotation of a circle ρα : R/Z

R/Z, x → x + α

2 Bernoulli shift σ : 2Z

2Z, σ(f)(n) = f(n + 1)

3 1+ : βZ

βZ, 1 + u = {1 + S, S ∈ u}

βZ is the ˇ Cech-Stone compactification on Z, i.e., the space of all ultrafilters on Z

Dana Bartoˇ sov´ a Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits an invariant probability measure

Dana Bartoˇ sov´ a Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits an invariant probability measure

1 Z Dana Bartoˇ sov´ a Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits an invariant probability measure

1 Z 2 locally compact groups Dana Bartoˇ sov´ a Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits an invariant probability measure

1 Z 2 locally compact groups 3 extremely amenable groups Dana Bartoˇ sov´ a Unique amenability of topological groups

Amenability

A topological group G is amenable if every G-flow admits an invariant probability measure

1 Z 2 locally compact groups 3 extremely amenable groups

G is extremely amenable if every G-flow has a fixed point

Dana Bartoˇ sov´ a Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow with a dense orbit admits a unique invariant probability measure.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow with a dense orbit admits a unique invariant probability measure.

1 precompact groups Dana Bartoˇ sov´ a Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow with a dense orbit admits a unique invariant probability measure.

1 precompact groups 2 ??? Dana Bartoˇ sov´ a Unique amenability of topological groups

Unique amenability

A topological group G is uniquely amenable if every G-flow with a dense orbit admits a unique invariant probability measure.

1 precompact groups 2 ???

A group is precompact if it is a subgroup of a compact group.

Dana Bartoˇ sov´ a Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a dense

• rbit that has two disjoint subflows.

Dana Bartoˇ sov´ a Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a dense

• rbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986) Dana Bartoˇ sov´ a Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a dense

• rbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986) 2 separable topological vector spaces (Ferri and Strauss,

2001)

Dana Bartoˇ sov´ a Unique amenability of topological groups

All non-examples thus far

Given a (amenable) topological group find a flow with a dense

• rbit that has two disjoint subflows.

1 locally compact groups (Lau and Paterson, 1986) 2 separable topological vector spaces (Ferri and Strauss,

2001)

3 groups of density < ℵω, automorphism groups, . . . (B.,

2013)

Dana Bartoˇ sov´ a Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X, x0) is a G-ambit if the orbit Gx0 is dense in X.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X, x0) is a G-ambit if the orbit Gx0 is dense in X. There is a compactification G ֒ → S(G) so that (S(G), e) is the greatest ambit, that is,

Dana Bartoˇ sov´ a Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X, x0) is a G-ambit if the orbit Gx0 is dense in X. There is a compactification G ֒ → S(G) so that (S(G), e) is the greatest ambit, that is, for every G-ambit (X, x0) there is a quotient map q : S(G)

X so that

Dana Bartoˇ sov´ a Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X, x0) is a G-ambit if the orbit Gx0 is dense in X. There is a compactification G ֒ → S(G) so that (S(G), e) is the greatest ambit, that is, for every G-ambit (X, x0) there is a quotient map q : S(G)

X so that

G × (X, x0) X

• G × (S(G), e)

G × (X, x0)

id×q

• G × (S(G), e)

S(G)

S(G)

X

id×q

• Dana Bartoˇ

sov´ a Unique amenability of topological groups

Universal flow with a dense orbit

A pointed G-flow (X, x0) is a G-ambit if the orbit Gx0 is dense in X. There is a compactification G ֒ → S(G) so that (S(G), e) is the greatest ambit, that is, for every G-ambit (X, x0) there is a quotient map q : S(G)

X so that

G × (X, x0) X

• G × (S(G), e)

G × (X, x0)

id×q

• G × (S(G), e)

S(G)

S(G)

X

id×q

• commutes and q(e) = x0.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it. In particular, if X has disjoint subflows, so does S(G).

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it. In particular, if X has disjoint subflows, so does S(G). Note, if X has disjoint flows than it has disjoint minimal flows, since

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it. In particular, if X has disjoint subflows, so does S(G). Note, if X has disjoint flows than it has disjoint minimal flows, since every flow has a minimal subflow.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it. In particular, if X has disjoint subflows, so does S(G). Note, if X has disjoint flows than it has disjoint minimal flows, since every flow has a minimal subflow. A G-flow is minimal if it has no proper non-empty closed invariant subset iff every orbit is dense.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) must be a witness

If X is a G-flow with a dense orbit, S(G) maps onto it. In particular, if X has disjoint subflows, so does S(G). Note, if X has disjoint flows than it has disjoint minimal flows, since every flow has a minimal subflow. A G-flow is minimal if it has no proper non-empty closed invariant subset iff every orbit is dense. Every minimal subflow of S(G) is the universal minimal flow.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology. (S(G), ·) is a right topological semigroup, i.e., · : S(G) × S(G)

S(G) is a semigroup operation and

·s : x → xs is continuous.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology. (S(G), ·) is a right topological semigroup, i.e., · : S(G) × S(G)

S(G) is a semigroup operation and

·s : x → xs is continuous. It extends the continuous action G × S(G)

S(G),

Dana Bartoˇ sov´ a Unique amenability of topological groups

Why do we like S(G)?

Algebra ♥ topology. (S(G), ·) is a right topological semigroup, i.e., · : S(G) × S(G)

S(G) is a semigroup operation and

·s : x → xs is continuous. It extends the continuous action G × S(G)

S(G),

which extends the continuous multiplication · : G × G

G.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Minimal subflows of S(G)

Let M be a minimal subflow of S(G) and m ∈ M. Then S(G)m = M, i.e., M is a (minimal) left ideal in S(G).

Dana Bartoˇ sov´ a Unique amenability of topological groups

Minimal subflows of S(G)

Let M be a minimal subflow of S(G) and m ∈ M. Then S(G)m = M, i.e., M is a (minimal) left ideal in S(G). Conversely, if N is a minimal left ideal of S(G) N is a subflow.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G gU = {gA : A ∈ U}

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G gU = {gA : A ∈ U} defines a continuous action G × βG

βG.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G gU = {gA : A ∈ U} defines a continuous action G × βG

βG.

Right multiplication (·, U) : G

βG

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G gU = {gA : A ∈ U} defines a continuous action G × βG

βG.

Right multiplication (·, U) : G

βG

can be continuously extended (·, U) : βG

βG.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G gU = {gA : A ∈ U} defines a continuous action G × βG

βG.

Right multiplication (·, U) : G

βG

can be continuously extended (·, U) : βG

βG.

UV = U − lim{gV : g ∈ G}

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G discrete

βG - ˇ Cech-Stone compactification of G gU = {gA : A ∈ U} defines a continuous action G × βG

βG.

Right multiplication (·, U) : G

βG

can be continuously extended (·, U) : βG

βG.

UV = U − lim{gV : g ∈ G} = {A ⊂ G : {g ∈ G : g−1A ∈ V} ∈ U}

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of open neighbourhoods of the identity.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of open neighbourhoods of the identity. A family F of subsets of G has the finite near intersection property (fnip) if

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of open neighbourhoods of the identity. A family F of subsets of G has the finite near intersection property (fnip) if for every F1, F2, . . . , Fn ∈ F and every V ∈ Ne(G)

n

• i=1

V Fi = ∅.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of open neighbourhoods of the identity. A family F of subsets of G has the finite near intersection property (fnip) if for every F1, F2, . . . , Fn ∈ F and every V ∈ Ne(G)

n

• i=1

V Fi = ∅. F is a near filter if it has fnip and is closed upwards.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near (ultra)filters

Let G be a topological group and Ne(G) a base of open neighbourhoods of the identity. A family F of subsets of G has the finite near intersection property (fnip) if for every F1, F2, . . . , Fn ∈ F and every V ∈ Ne(G)

n

• i=1

V Fi = ∅. F is a near filter if it has fnip and is closed upwards. F is a near ultrafilter if it a maximal near filter.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets ˆ A = {U ∈ S(G) : A ∈ U} for A ⊂ G.

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets ˆ A = {U ∈ S(G) : A ∈ U} for A ⊂ G. The action G × S(G)

S(G), gU = {gU : U ∈ U}

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets ˆ A = {U ∈ S(G) : A ∈ U} for A ⊂ G. The action G × S(G)

S(G), gU = {gU : U ∈ U}

Extends to S(G) × S(G)

S(G)

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets ˆ A = {U ∈ S(G) : A ∈ U} for A ⊂ G. The action G × S(G)

S(G), gU = {gU : U ∈ U}

Extends to S(G) × S(G)

S(G)

UV = U − lim{gV : g ∈ G}

Dana Bartoˇ sov´ a Unique amenability of topological groups

S(G) for G topological

S(G) = the space of near ultrafilters with base for closed sets ˆ A = {U ∈ S(G) : A ∈ U} for A ⊂ G. The action G × S(G)

S(G), gU = {gU : U ∈ U}

Extends to S(G) × S(G)

S(G)

UV = U − lim{gV : g ∈ G} = {A ⊂ G : V {g ∈ G : g−1A ∈ V} ∈ U, V ∈ Ne(G)}

Dana Bartoˇ sov´ a Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if T

βG contains a minimal left ideal of

βG.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if T

βG contains a minimal left ideal of

βG. Equivalently, {gT : g ∈ G} has the finite instersection property.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if T

βG contains a minimal left ideal of

βG. Equivalently, {gT : g ∈ G} has the finite instersection property. Equivalently, for every finite F ⊂ G there is x ∈ G such that Fx ⊂ G.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Thick sets

A subset T ⊆ G is thick if T

βG contains a minimal left ideal of

βG. Equivalently, {gT : g ∈ G} has the finite instersection property. Equivalently, for every finite F ⊂ G there is x ∈ G such that Fx ⊂ G. Minimal left ideals ≡ maximal filters of thick sets.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Prethick sets

A subset T ⊆ G is prethick if T

S(G) contains a minimal left

ideal in S(G).

Dana Bartoˇ sov´ a Unique amenability of topological groups

Prethick sets

A subset T ⊆ G is prethick if T

S(G) contains a minimal left

ideal in S(G). Equivalently, V T is thick for every open V ∈ Ne(G).

Dana Bartoˇ sov´ a Unique amenability of topological groups

Prethick sets

A subset T ⊆ G is prethick if T

S(G) contains a minimal left

ideal in S(G). Equivalently, V T is thick for every open V ∈ Ne(G). Minimal left ideals ≡ maximal near filters of prethick sets.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Disjoint thick sets

Theorem (Carlson, Hindman, McLeod, Strauss (2008)) Discrete G of cardinality κ can be split into κ disjoint thick sets.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Disjoint thick sets

Theorem (Carlson, Hindman, McLeod, Strauss (2008)) Discrete G of cardinality κ can be split into κ disjoint thick sets. Corollary If G is as above, βG contains 22κ disjoint minimal left ideals.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Disjoint thick sets

Theorem (Carlson, Hindman, McLeod, Strauss (2008)) Discrete G of cardinality κ can be split into κ disjoint thick sets. Corollary If G is as above, βG contains 22κ disjoint minimal left ideals. CONSTRUCTION

1 enumerate finite subsets of G into (Fλ)λ<κ. 2 recursively pick (xλ)λ>κ so that Fλxλ ∩

µ<λ Fµxµ = ∅.

3 κ = ˙

• ν<κIν, so that for any ν every finite set F ⊂ G is

contained in Fλ for some λ ∈ Iν.

4 Tν =

λ∈Iν Fλxλ for ν < κ are disjoint thick.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near disjoint thick sets

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}. Theorem (B., 2013) If there is V ∈ Ne(G) so that κ(V ) = κ(V 6) ≥ ℵ0 then G contains κ(V ) thick sets whose V -saturations are disjoint.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Near disjoint thick sets

For V ∈ Ne(G), set κ(V ) = min{|Γ| : ∀g, h ∈ Γ, gV ∩ hV = ∅ & ΓV2 = G}. Theorem (B., 2013) If there is V ∈ Ne(G) so that κ(V ) = κ(V 6) ≥ ℵ0 then G contains κ(V ) thick sets whose V -saturations are disjoint. Corollary If G is as above, S(G) contains 22κ(V ) disjoint minimal left ideals.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)

satisfies κ(V ) = κ(V 6) ≥ ℵ0?

Dana Bartoˇ sov´ a Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)

satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimal

left ideal in S(G)?

Dana Bartoˇ sov´ a Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)

satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimal

left ideal in S(G)?

3 Is there a non-precompact extremely amenable group G

with a single minimal left ideal in S(G)?

Dana Bartoˇ sov´ a Unique amenability of topological groups

Open

1 Is there a topological group G such that no V ∈ Ne(G)

satisfies κ(V ) = κ(V 6) ≥ ℵ0?

2 Is there a non-precompact group G with a single minimal

left ideal in S(G)?

3 Is there a non-precompact extremely amenable group G

with a single minimal left ideal in S(G)?

4 Is there a non-precompact group G that is uniquely

amenable.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·)

Dana Bartoˇ sov´ a Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·) = minimal left ideals of S(G)

Dana Bartoˇ sov´ a Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·) = minimal left ideals of S(G) If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998) Dana Bartoˇ sov´ a Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·) = minimal left ideals of S(G) If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998) 2 G can be algebraically embdedded into a compact group.

(Zelenyuk, 2009)

Dana Bartoˇ sov´ a Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·) = minimal left ideals of S(G) If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998) 2 G can be algebraically embdedded into a compact group.

(Zelenyuk, 2009) It means that S(G) need to contain infinitely many minimal left ideals.

Dana Bartoˇ sov´ a Unique amenability of topological groups

Another point of attack

K(S(G)) = minimal (both-sided) ideal in (S(G), ·) = minimal left ideals of S(G) If G is discrete than K(βG) is not closed when

1 G is countable. (Hindman and Strauss, 1998) 2 G can be algebraically embdedded into a compact group.

(Zelenyuk, 2009) It means that S(G) need to contain infinitely many minimal left ideals. Theorem (Hindman and Strauss, 2017) K(βN) is not Borel.

Dana Bartoˇ sov´ a Unique amenability of topological groups

G Polish

Theorem (B. and Zucker 2017) K(S(G)) is closed if and only if minimal left ideals of S(G) are metrizable.

Dana Bartoˇ sov´ a Unique amenability of topological groups

The end

THANK YOU!

Dana Bartoˇ sov´ a Unique amenability of topological groups