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 � X - a continuous action G × X Dana Bartoˇ sov´ a Unique amenability of topological groups

G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space Dana Bartoˇ sov´ a Unique amenability of topological groups

G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x Dana Bartoˇ sov´ a Unique amenability of topological groups

G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x � X - a homeomorphism ( g, · ) : X Dana Bartoˇ sov´ a Unique amenability of topological groups

G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x � X - a homeomorphism ( g, · ) : X � Homeo(X) - a continuous homomorphism G Dana Bartoˇ sov´ a Unique amenability of topological groups

G -flow � X - a continuous action G × X ↑ ↑ topological compact group Hausdorff space g ( hx ) = ( gh ) x ex = x � X - a homeomorphism ( g, · ) : X � Homeo(X) - a continuous homomorphism G We call X a G -flow. Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples � X homeomorphism, X compact Hausdorff f : X Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples � X homeomorphism, X compact Hausdorff f : X gives rise to a Z -flow � X, ( n, x ) �→ f z ( x ) Z × X Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples � X homeomorphism, X compact Hausdorff f : X gives rise to a Z -flow � X, ( n, x ) �→ f z ( x ) Z × X 1 rotation of a circle ρ α : R / Z � R / Z , x �→ x + α Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples � X homeomorphism, X compact Hausdorff f : X gives rise to a Z -flow � X, ( n, x ) �→ f z ( x ) Z × X 1 rotation of a circle ρ α : R / Z � R / Z , x �→ x + α � 2 Z , σ ( f )( n ) = f ( n + 1) 2 Bernoulli shift σ : 2 Z Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples � X homeomorphism, X compact Hausdorff f : X gives rise to a Z -flow � X, ( n, x ) �→ f z ( x ) Z × X 1 rotation of a circle ρ α : R / Z � R / Z , x �→ x + α � 2 Z , σ ( f )( n ) = f ( n + 1) 2 Bernoulli shift σ : 2 Z � β Z , 1 + u = { 1 + S, S ∈ u } 3 1+ : β Z Dana Bartoˇ sov´ a Unique amenability of topological groups

Examples � X homeomorphism, X compact Hausdorff f : X gives rise to a Z -flow � X, ( n, x ) �→ f z ( x ) Z × X 1 rotation of a circle ρ α : R / Z � R / Z , x �→ x + α � 2 Z , σ ( f )( n ) = f ( n + 1) 2 Bernoulli shift σ : 2 Z � β Z , 1 + u = { 1 + S, S ∈ u } 3 1+ : β Z 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 orbit 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 orbit 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 orbit 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 orbit 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 < ℵ ω , automorphism groups, . . . (B., 2013) Dana Bartoˇ sov´ a Unique amenability of topological groups

Universal flow with a dense orbit A pointed G -flow ( X, x 0 ) is a G -ambit if the orbit Gx 0 is dense in X. Dana Bartoˇ sov´ a Unique amenability of topological groups

Universal flow with a dense orbit A pointed G -flow ( X, x 0 ) is a G -ambit if the orbit Gx 0 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, x 0 ) is a G -ambit if the orbit Gx 0 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, x 0 ) there is a � X so that quotient map q : S ( G ) Dana Bartoˇ sov´ a Unique amenability of topological groups

� � � Universal flow with a dense orbit A pointed G -flow ( X, x 0 ) is a G -ambit if the orbit Gx 0 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, x 0 ) there is a � X so that quotient map q : S ( G ) � S ( G ) G × ( S ( G ) , e ) G × ( S ( G ) , e ) S ( G ) id × q id × q G × ( X, x 0 ) G × ( X, x 0 ) X X Dana Bartoˇ sov´ a Unique amenability of topological groups

� � � Universal flow with a dense orbit A pointed G -flow ( X, x 0 ) is a G -ambit if the orbit Gx 0 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, x 0 ) there is a � X so that quotient map q : S ( G ) � S ( G ) G × ( S ( G ) , e ) G × ( S ( G ) , e ) S ( G ) id × q id × q G × ( X, x 0 ) G × ( X, x 0 ) X X commutes and q ( e ) = x 0 . 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