Property (T) for quantum groups from the Property (T) for groups - PowerPoint PPT Presentation

Property (T) for quantum groups from the dual point of view David Kyed Property (T) for quantum groups from the Property (T) for groups dual point of view Quantum groups Property (T) for quantum groups David Kyed The dual picture Georg-August-Universit¨ at G¨ ottingen Copenhagen February 2010

Property (T) Rough outline for quantum groups from the dual point of view David Kyed Property (T) for groups Quantum groups Property (T) • Property (T) for groups for quantum groups • Compact and discrete quantum groups The dual picture • Property (T) for quantum groups • Different characterizations of property (T)

Property (T) Property (T) for groups for quantum groups from the dual point of view David Kyed We start with the following data: Property (T) • Γ discrete, countable group, for groups Quantum • π : Γ → U ( H ) a representation, groups • E ⊆ Γ finite, δ > 0 and ξ ∈ ( H ) 1 . Property (T) for quantum groups Then The dual • ξ is called ( E , δ )-invariant if � π ( γ ) ξ − ξ � < δ for γ ∈ E . picture • π is said to have almost invariant vectors if such ξ exists for all E and δ . • Γ is said to have Kazhdan’s property (T) if every π that has almost invariant vectors actually has a non-zero invariant vector. Property (T) is of importance in many fields — in particular operator algebras.

Property (T) Property (T) for groups for quantum groups from the dual point of view David Kyed We start with the following data: Property (T) • Γ discrete, countable group, for groups Quantum • π : Γ → U ( H ) a representation, groups • E ⊆ Γ finite, δ > 0 and ξ ∈ ( H ) 1 . Property (T) for quantum groups Then The dual • ξ is called ( E , δ )-invariant if � π ( γ ) ξ − ξ � < δ for γ ∈ E . picture • π is said to have almost invariant vectors if such ξ exists for all E and δ . • Γ is said to have Kazhdan’s property (T) if every π that has almost invariant vectors actually has a non-zero invariant vector. Property (T) is of importance in many fields — in particular operator algebras.

Property (T) Property (T) for groups for quantum groups from the dual point of view David Kyed We start with the following data: Property (T) • Γ discrete, countable group, for groups Quantum • π : Γ → U ( H ) a representation, groups • E ⊆ Γ finite, δ > 0 and ξ ∈ ( H ) 1 . Property (T) for quantum groups Then The dual • ξ is called ( E , δ )-invariant if � π ( γ ) ξ − ξ � < δ for γ ∈ E . picture • π is said to have almost invariant vectors if such ξ exists for all E and δ . • Γ is said to have Kazhdan’s property (T) if every π that has almost invariant vectors actually has a non-zero invariant vector. Property (T) is of importance in many fields — in particular operator algebras.

Property (T) Property (T) for groups for quantum groups from the dual point of view David Kyed We start with the following data: Property (T) • Γ discrete, countable group, for groups Quantum • π : Γ → U ( H ) a representation, groups • E ⊆ Γ finite, δ > 0 and ξ ∈ ( H ) 1 . Property (T) for quantum groups Then The dual • ξ is called ( E , δ )-invariant if � π ( γ ) ξ − ξ � < δ for γ ∈ E . picture • π is said to have almost invariant vectors if such ξ exists for all E and δ . • Γ is said to have Kazhdan’s property (T) if every π that has almost invariant vectors actually has a non-zero invariant vector. Property (T) is of importance in many fields — in particular operator algebras.

Property (T) Property (T) for groups for quantum groups from the dual point of view David Kyed We start with the following data: Property (T) • Γ discrete, countable group, for groups Quantum • π : Γ → U ( H ) a representation, groups • E ⊆ Γ finite, δ > 0 and ξ ∈ ( H ) 1 . Property (T) for quantum groups Then The dual • ξ is called ( E , δ )-invariant if � π ( γ ) ξ − ξ � < δ for γ ∈ E . picture • π is said to have almost invariant vectors if such ξ exists for all E and δ . • Γ is said to have Kazhdan’s property (T) if every π that has almost invariant vectors actually has a non-zero invariant vector. Property (T) is of importance in many fields — in particular operator algebras.

Property (T) for quantum groups from There are many ways to describe property (T). Here are two: the dual point of view Theorem (Delorme-Guichardet, de la Harpe-Valette) David Kyed Property (T) The following conditions are equivalent for groups • Γ has property (T) . Quantum groups • Any sequence ϕ n : Γ → C of normalized, positive definite Property (T) for quantum functions converging pointwise to 1 has to converge groups uniformly. The dual picture • For any π : Γ → U ( H ) we have H 1 (Γ , H ) = 0 . i.e. ϕ n ( e ) = 1 and ϕ n ( x ∗ x ) ≥ 0 for x ∈ C Γ. First group cohomology of Γ=first Hochschild cohomology H 1 ( C Γ , π H ε ). The aim of the talk is to discuss a similar result for quantum groups. We first introduce these objects:

Property (T) for quantum groups from There are many ways to describe property (T). Here are two: the dual point of view Theorem (Delorme-Guichardet, de la Harpe-Valette) David Kyed Property (T) The following conditions are equivalent for groups • Γ has property (T) . Quantum groups • Any sequence ϕ n : Γ → C of normalized, positive definite Property (T) for quantum functions converging pointwise to 1 has to converge groups uniformly. The dual picture • For any π : Γ → U ( H ) we have H 1 (Γ , H ) = 0 . i.e. ϕ n ( e ) = 1 and ϕ n ( x ∗ x ) ≥ 0 for x ∈ C Γ. First group cohomology of Γ=first Hochschild cohomology H 1 ( C Γ , π H ε ). The aim of the talk is to discuss a similar result for quantum groups. We first introduce these objects:

Property (T) for quantum groups from There are many ways to describe property (T). Here are two: the dual point of view Theorem (Delorme-Guichardet, de la Harpe-Valette) David Kyed Property (T) The following conditions are equivalent for groups • Γ has property (T) . Quantum groups • Any sequence ϕ n : Γ → C of normalized, positive definite Property (T) for quantum functions converging pointwise to 1 has to converge groups uniformly. The dual picture • For any π : Γ → U ( H ) we have H 1 (Γ , H ) = 0 . i.e. ϕ n ( e ) = 1 and ϕ n ( x ∗ x ) ≥ 0 for x ∈ C Γ. First group cohomology of Γ=first Hochschild cohomology H 1 ( C Γ , π H ε ). The aim of the talk is to discuss a similar result for quantum groups. We first introduce these objects:

Property (T) for quantum groups from There are many ways to describe property (T). Here are two: the dual point of view Theorem (Delorme-Guichardet, de la Harpe-Valette) David Kyed Property (T) The following conditions are equivalent for groups • Γ has property (T) . Quantum groups • Any sequence ϕ n : Γ → C of normalized, positive definite Property (T) for quantum functions converging pointwise to 1 has to converge groups uniformly. The dual picture • For any π : Γ → U ( H ) we have H 1 (Γ , H ) = 0 . i.e. ϕ n ( e ) = 1 and ϕ n ( x ∗ x ) ≥ 0 for x ∈ C Γ. First group cohomology of Γ=first Hochschild cohomology H 1 ( C Γ , π H ε ). The aim of the talk is to discuss a similar result for quantum groups. We first introduce these objects:

Property (T) Compact quantum groups for quantum groups from the dual point of view Definition (Woronowicz) David Kyed A compact quantum group G consists of a unital, separable Property (T) for groups C ∗ -algebra C ( G ) together with a unital ∗ -homomorphism Quantum ∆: C ( G ) → C ( G ) ⊗ C ( G ) satisfying groups Property (T) • (id ⊗ ∆)∆ = (∆ ⊗ id)∆ , for quantum groups • a certain density condition. The dual picture Example: C ( G ) with G compact group and ∆( f )( s , t ) = f ( st ). Example: C ∗ red (Γ) with Γ discrete and ∆ γ = γ ⊗ γ . Theorem (Woronowicz) Any compact quantum group with C ( G ) abelian is of the form C ( G ) for an honest compact group G. As groups have representations, quantum groups have corepresentations:

Property (T) Compact quantum groups for quantum groups from the dual point of view Definition (Woronowicz) David Kyed A compact quantum group G consists of a unital, separable Property (T) for groups C ∗ -algebra C ( G ) together with a unital ∗ -homomorphism Quantum ∆: C ( G ) → C ( G ) ⊗ C ( G ) satisfying groups Property (T) • (id ⊗ ∆)∆ = (∆ ⊗ id)∆ , for quantum groups • a certain density condition. The dual picture Example: C ( G ) with G compact group and ∆( f )( s , t ) = f ( st ). Example: C ∗ red (Γ) with Γ discrete and ∆ γ = γ ⊗ γ . Theorem (Woronowicz) Any compact quantum group with C ( G ) abelian is of the form C ( G ) for an honest compact group G. As groups have representations, quantum groups have corepresentations: