![Page 1: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/1.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Property T for locally compact quantum groups
Chi-Keung Ng
The Chern Institute of MathematicsNankai University
Workshop on Operator Spaces,Locally Compact Quantum Groups and Amenability
Fields Institute; May 26-30, 2014
(joint works with Xiao Chen and Anthony To Ming Lau)
Ng Property T for locally compact quantum groups
![Page 2: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/2.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Property T for locally compact groups was first introduced byD. Kazhdan in the 1960s. He used it to show that some latticesare finitely generated. This notion was proved to be very usefuland has many equivalent formulations.
• P. Fima partially extended the notion of property T to discretequantum groups and he showed that discrete quantum groupswith property T are of Kac type.• D. Kyed and M. Soltan partially generalize some equivalencesof property T to discrete quantum groups.• D. Kyed obtained a Delorme-Guichardet type theorem (acohomology description for property T ) for discrete quantumgroups.• M. Daws, P. Fima, A. Skalski and S. White gave the definitionof property T for general locally compact quantum groups, intheir paper on the Haagerup property.
Ng Property T for locally compact quantum groups
![Page 3: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/3.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Property T for locally compact groups was first introduced byD. Kazhdan in the 1960s. He used it to show that some latticesare finitely generated. This notion was proved to be very usefuland has many equivalent formulations.• P. Fima partially extended the notion of property T to discretequantum groups and he showed that discrete quantum groupswith property T are of Kac type.
• D. Kyed and M. Soltan partially generalize some equivalencesof property T to discrete quantum groups.• D. Kyed obtained a Delorme-Guichardet type theorem (acohomology description for property T ) for discrete quantumgroups.• M. Daws, P. Fima, A. Skalski and S. White gave the definitionof property T for general locally compact quantum groups, intheir paper on the Haagerup property.
Ng Property T for locally compact quantum groups
![Page 4: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/4.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Property T for locally compact groups was first introduced byD. Kazhdan in the 1960s. He used it to show that some latticesare finitely generated. This notion was proved to be very usefuland has many equivalent formulations.• P. Fima partially extended the notion of property T to discretequantum groups and he showed that discrete quantum groupswith property T are of Kac type.• D. Kyed and M. Soltan partially generalize some equivalencesof property T to discrete quantum groups.
• D. Kyed obtained a Delorme-Guichardet type theorem (acohomology description for property T ) for discrete quantumgroups.• M. Daws, P. Fima, A. Skalski and S. White gave the definitionof property T for general locally compact quantum groups, intheir paper on the Haagerup property.
Ng Property T for locally compact quantum groups
![Page 5: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/5.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Property T for locally compact groups was first introduced byD. Kazhdan in the 1960s. He used it to show that some latticesare finitely generated. This notion was proved to be very usefuland has many equivalent formulations.• P. Fima partially extended the notion of property T to discretequantum groups and he showed that discrete quantum groupswith property T are of Kac type.• D. Kyed and M. Soltan partially generalize some equivalencesof property T to discrete quantum groups.• D. Kyed obtained a Delorme-Guichardet type theorem (acohomology description for property T ) for discrete quantumgroups.
• M. Daws, P. Fima, A. Skalski and S. White gave the definitionof property T for general locally compact quantum groups, intheir paper on the Haagerup property.
Ng Property T for locally compact quantum groups
![Page 6: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/6.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Property T for locally compact groups was first introduced byD. Kazhdan in the 1960s. He used it to show that some latticesare finitely generated. This notion was proved to be very usefuland has many equivalent formulations.• P. Fima partially extended the notion of property T to discretequantum groups and he showed that discrete quantum groupswith property T are of Kac type.• D. Kyed and M. Soltan partially generalize some equivalencesof property T to discrete quantum groups.• D. Kyed obtained a Delorme-Guichardet type theorem (acohomology description for property T ) for discrete quantumgroups.• M. Daws, P. Fima, A. Skalski and S. White gave the definitionof property T for general locally compact quantum groups, intheir paper on the Haagerup property.
Ng Property T for locally compact quantum groups
![Page 7: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/7.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• If G is a locally compact group and µ : G→ L(H) is acontinuous unitary representation, then a net {ξi}i∈I of unitvectors in H is called an almost invariant unit vector (for µ) if
supt∈K‖µt (ξi)− ξi‖ → 0,
for any compact subset K ⊆ G.
•We denote by C∗(G) the full group C∗-algebra of G and by πµthe ∗-representation of C∗(G) induced by µ.• 1G is the trivial one-dimensional representation of G.• G is the topological space of irreducible unitaryrepresentations of G.
Ng Property T for locally compact quantum groups
![Page 8: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/8.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• If G is a locally compact group and µ : G→ L(H) is acontinuous unitary representation, then a net {ξi}i∈I of unitvectors in H is called an almost invariant unit vector (for µ) if
supt∈K‖µt (ξi)− ξi‖ → 0,
for any compact subset K ⊆ G.•We denote by C∗(G) the full group C∗-algebra of G and by πµthe ∗-representation of C∗(G) induced by µ.
• 1G is the trivial one-dimensional representation of G.• G is the topological space of irreducible unitaryrepresentations of G.
Ng Property T for locally compact quantum groups
![Page 9: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/9.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• If G is a locally compact group and µ : G→ L(H) is acontinuous unitary representation, then a net {ξi}i∈I of unitvectors in H is called an almost invariant unit vector (for µ) if
supt∈K‖µt (ξi)− ξi‖ → 0,
for any compact subset K ⊆ G.•We denote by C∗(G) the full group C∗-algebra of G and by πµthe ∗-representation of C∗(G) induced by µ.• 1G is the trivial one-dimensional representation of G.
• G is the topological space of irreducible unitaryrepresentations of G.
Ng Property T for locally compact quantum groups
![Page 10: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/10.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• If G is a locally compact group and µ : G→ L(H) is acontinuous unitary representation, then a net {ξi}i∈I of unitvectors in H is called an almost invariant unit vector (for µ) if
supt∈K‖µt (ξi)− ξi‖ → 0,
for any compact subset K ⊆ G.•We denote by C∗(G) the full group C∗-algebra of G and by πµthe ∗-representation of C∗(G) induced by µ.• 1G is the trivial one-dimensional representation of G.• G is the topological space of irreducible unitaryrepresentations of G.
Ng Property T for locally compact quantum groups
![Page 11: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/11.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G is said to have property T if any continuous unitaryrepresentation of G that has an almost invariant unit vectoractually has an invariant unit vector.
• Property T of G is equivalent to:(T1) C∗(G) ∼= kerπ1G ⊕ C canonically.(T2) ∃ minimal projection p ∈ M(C∗(G)) such that π1G (p) = 1.
(T3) 1G is an isolated point in G.(T4) All fin. dim. representations in G are isolated points.(T5) ∃ a fin. dim. representation in G which is an isolated point.• The equivalences of prop. T with T1-T5 for discrete quantumgroups are some of the main results in the paper by Kyed andSoltan.• (Delorme-Guichardet theorem) If G is σ-compact, then it hasproperty T if and only if H1(G, µ) = (0) for any continuousunitary representation µ of G.
Ng Property T for locally compact quantum groups
![Page 12: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/12.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G is said to have property T if any continuous unitaryrepresentation of G that has an almost invariant unit vectoractually has an invariant unit vector.• Property T of G is equivalent to:(T1) C∗(G) ∼= kerπ1G ⊕ C canonically.(T2) ∃ minimal projection p ∈ M(C∗(G)) such that π1G (p) = 1.
(T3) 1G is an isolated point in G.(T4) All fin. dim. representations in G are isolated points.(T5) ∃ a fin. dim. representation in G which is an isolated point.
• The equivalences of prop. T with T1-T5 for discrete quantumgroups are some of the main results in the paper by Kyed andSoltan.• (Delorme-Guichardet theorem) If G is σ-compact, then it hasproperty T if and only if H1(G, µ) = (0) for any continuousunitary representation µ of G.
Ng Property T for locally compact quantum groups
![Page 13: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/13.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G is said to have property T if any continuous unitaryrepresentation of G that has an almost invariant unit vectoractually has an invariant unit vector.• Property T of G is equivalent to:(T1) C∗(G) ∼= kerπ1G ⊕ C canonically.(T2) ∃ minimal projection p ∈ M(C∗(G)) such that π1G (p) = 1.
(T3) 1G is an isolated point in G.(T4) All fin. dim. representations in G are isolated points.(T5) ∃ a fin. dim. representation in G which is an isolated point.• The equivalences of prop. T with T1-T5 for discrete quantumgroups are some of the main results in the paper by Kyed andSoltan.
• (Delorme-Guichardet theorem) If G is σ-compact, then it hasproperty T if and only if H1(G, µ) = (0) for any continuousunitary representation µ of G.
Ng Property T for locally compact quantum groups
![Page 14: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/14.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G is said to have property T if any continuous unitaryrepresentation of G that has an almost invariant unit vectoractually has an invariant unit vector.• Property T of G is equivalent to:(T1) C∗(G) ∼= kerπ1G ⊕ C canonically.(T2) ∃ minimal projection p ∈ M(C∗(G)) such that π1G (p) = 1.
(T3) 1G is an isolated point in G.(T4) All fin. dim. representations in G are isolated points.(T5) ∃ a fin. dim. representation in G which is an isolated point.• The equivalences of prop. T with T1-T5 for discrete quantumgroups are some of the main results in the paper by Kyed andSoltan.• (Delorme-Guichardet theorem) If G is σ-compact, then it hasproperty T if and only if H1(G, µ) = (0) for any continuousunitary representation µ of G.
Ng Property T for locally compact quantum groups
![Page 15: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/15.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Some notations and prelimiaries concerning a C∗-algebra A:• Rep(A): unitary equiv. classes of non-deg. ∗-rep. of A
• If (µ,H), (ν,K) ∈ Rep(A), we set* µ ⊂ ν if ∃ isom. V : H→ K s.t. µ(a) = V ∗ν(a)V , ∀a ∈ A;* µ ≺ ν if ker ν ⊂ kerµ.• A ⊆ Rep(A): irred. rep. (equip with Fell top.)
Lemma
Let A be a C∗-algebra and (µ,H) ∈ A.(a) The following statements are equivalent.
1 µ is an isolated point in A.2 ∀(π,K) ∈ Rep(A) with µ ≺ π, one has µ ⊂ π.3 A = kerµ⊕
⋂ν∈A\{µ} ker ν.
(b) If dimH <∞, then {µ} is a closed subset of A.
Ng Property T for locally compact quantum groups
![Page 16: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/16.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Some notations and prelimiaries concerning a C∗-algebra A:• Rep(A): unitary equiv. classes of non-deg. ∗-rep. of A• If (µ,H), (ν,K) ∈ Rep(A), we set
* µ ⊂ ν if ∃ isom. V : H→ K s.t. µ(a) = V ∗ν(a)V , ∀a ∈ A;* µ ≺ ν if ker ν ⊂ kerµ.
• A ⊆ Rep(A): irred. rep. (equip with Fell top.)
Lemma
Let A be a C∗-algebra and (µ,H) ∈ A.(a) The following statements are equivalent.
1 µ is an isolated point in A.2 ∀(π,K) ∈ Rep(A) with µ ≺ π, one has µ ⊂ π.3 A = kerµ⊕
⋂ν∈A\{µ} ker ν.
(b) If dimH <∞, then {µ} is a closed subset of A.
Ng Property T for locally compact quantum groups
![Page 17: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/17.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Some notations and prelimiaries concerning a C∗-algebra A:• Rep(A): unitary equiv. classes of non-deg. ∗-rep. of A• If (µ,H), (ν,K) ∈ Rep(A), we set
* µ ⊂ ν if ∃ isom. V : H→ K s.t. µ(a) = V ∗ν(a)V , ∀a ∈ A;* µ ≺ ν if ker ν ⊂ kerµ.• A ⊆ Rep(A): irred. rep. (equip with Fell top.)
Lemma
Let A be a C∗-algebra and (µ,H) ∈ A.(a) The following statements are equivalent.
1 µ is an isolated point in A.2 ∀(π,K) ∈ Rep(A) with µ ≺ π, one has µ ⊂ π.3 A = kerµ⊕
⋂ν∈A\{µ} ker ν.
(b) If dimH <∞, then {µ} is a closed subset of A.
Ng Property T for locally compact quantum groups
![Page 18: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/18.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Some notations and prelimiaries concerning a C∗-algebra A:• Rep(A): unitary equiv. classes of non-deg. ∗-rep. of A• If (µ,H), (ν,K) ∈ Rep(A), we set
* µ ⊂ ν if ∃ isom. V : H→ K s.t. µ(a) = V ∗ν(a)V , ∀a ∈ A;* µ ≺ ν if ker ν ⊂ kerµ.• A ⊆ Rep(A): irred. rep. (equip with Fell top.)
Lemma
Let A be a C∗-algebra and (µ,H) ∈ A.(a) The following statements are equivalent.
1 µ is an isolated point in A.2 ∀(π,K) ∈ Rep(A) with µ ≺ π, one has µ ⊂ π.3 A = kerµ⊕
⋂ν∈A\{µ} ker ν.
(b) If dimH <∞, then {µ} is a closed subset of A.
Ng Property T for locally compact quantum groups
![Page 19: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/19.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Notations and prelimiaries on loc. comp. quant. groups:• G: a loc. comp. quant. group with reduced C∗-algebraicpresentation (C0(G),∆, ϕ, ψ).
• U ∈ M(K(H)⊗ C0(G)) is called a unitary corepresentation ofC0(G) if (id⊗∆)(U) = U12U13.•We denote by 1G ∈ M(C0(G)) the identity (which is the trivial1-dim. unit. corepn.)• (Cu
0(G), ∆u): the universal Hopf C∗-alg. associate with thedual group G of G.• V u
G ∈ M(Cu0(G)⊗ C0(G)): unitary that implements the bij.
corresp. between unitary coreprens. of C0(G) and non-deg.∗-repns. of Cu
0(G) through U = (πU ⊗ id)(V uG).
Ng Property T for locally compact quantum groups
![Page 20: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/20.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Notations and prelimiaries on loc. comp. quant. groups:• G: a loc. comp. quant. group with reduced C∗-algebraicpresentation (C0(G),∆, ϕ, ψ).• U ∈ M(K(H)⊗ C0(G)) is called a unitary corepresentation ofC0(G) if (id⊗∆)(U) = U12U13.
•We denote by 1G ∈ M(C0(G)) the identity (which is the trivial1-dim. unit. corepn.)• (Cu
0(G), ∆u): the universal Hopf C∗-alg. associate with thedual group G of G.• V u
G ∈ M(Cu0(G)⊗ C0(G)): unitary that implements the bij.
corresp. between unitary coreprens. of C0(G) and non-deg.∗-repns. of Cu
0(G) through U = (πU ⊗ id)(V uG).
Ng Property T for locally compact quantum groups
![Page 21: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/21.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Notations and prelimiaries on loc. comp. quant. groups:• G: a loc. comp. quant. group with reduced C∗-algebraicpresentation (C0(G),∆, ϕ, ψ).• U ∈ M(K(H)⊗ C0(G)) is called a unitary corepresentation ofC0(G) if (id⊗∆)(U) = U12U13.•We denote by 1G ∈ M(C0(G)) the identity (which is the trivial1-dim. unit. corepn.)
• (Cu0(G), ∆u): the universal Hopf C∗-alg. associate with the
dual group G of G.• V u
G ∈ M(Cu0(G)⊗ C0(G)): unitary that implements the bij.
corresp. between unitary coreprens. of C0(G) and non-deg.∗-repns. of Cu
0(G) through U = (πU ⊗ id)(V uG).
Ng Property T for locally compact quantum groups
![Page 22: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/22.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Notations and prelimiaries on loc. comp. quant. groups:• G: a loc. comp. quant. group with reduced C∗-algebraicpresentation (C0(G),∆, ϕ, ψ).• U ∈ M(K(H)⊗ C0(G)) is called a unitary corepresentation ofC0(G) if (id⊗∆)(U) = U12U13.•We denote by 1G ∈ M(C0(G)) the identity (which is the trivial1-dim. unit. corepn.)• (Cu
0(G), ∆u): the universal Hopf C∗-alg. associate with thedual group G of G.
• V uG ∈ M(Cu
0(G)⊗ C0(G)): unitary that implements the bij.corresp. between unitary coreprens. of C0(G) and non-deg.∗-repns. of Cu
0(G) through U = (πU ⊗ id)(V uG).
Ng Property T for locally compact quantum groups
![Page 23: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/23.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Notations and prelimiaries on loc. comp. quant. groups:• G: a loc. comp. quant. group with reduced C∗-algebraicpresentation (C0(G),∆, ϕ, ψ).• U ∈ M(K(H)⊗ C0(G)) is called a unitary corepresentation ofC0(G) if (id⊗∆)(U) = U12U13.•We denote by 1G ∈ M(C0(G)) the identity (which is the trivial1-dim. unit. corepn.)• (Cu
0(G), ∆u): the universal Hopf C∗-alg. associate with thedual group G of G.• V u
G ∈ M(Cu0(G)⊗ C0(G)): unitary that implements the bij.
corresp. between unitary coreprens. of C0(G) and non-deg.∗-repns. of Cu
0(G) through U = (πU ⊗ id)(V uG).
Ng Property T for locally compact quantum groups
![Page 24: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/24.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• (λG,L2(G)) is the GNS representation for ϕ.
• U ∈ M(K(H)⊗ C0(G)): unitary corepn. of C0(G).(a) ξ ∈ H is s.t.b. a U-invariant if U(ξ ⊗ η) = ξ ⊗ η, ∀η ∈ L2(G).(b) A net {ξi}i∈I of unit vectors in H is called an almostU-invariant unit vector if ‖U(ξi ⊗ η)− ξi ⊗ η‖ → 0, ∀η ∈ L2(G).
• U has a non-zero invariant vector⇔ π1G ⊂ πU .
• U has almost invariant vectors⇔ π1G ≺ πU .
Ng Property T for locally compact quantum groups
![Page 25: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/25.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• (λG,L2(G)) is the GNS representation for ϕ.
• U ∈ M(K(H)⊗ C0(G)): unitary corepn. of C0(G).(a) ξ ∈ H is s.t.b. a U-invariant if U(ξ ⊗ η) = ξ ⊗ η, ∀η ∈ L2(G).
(b) A net {ξi}i∈I of unit vectors in H is called an almostU-invariant unit vector if ‖U(ξi ⊗ η)− ξi ⊗ η‖ → 0, ∀η ∈ L2(G).
• U has a non-zero invariant vector⇔ π1G ⊂ πU .
• U has almost invariant vectors⇔ π1G ≺ πU .
Ng Property T for locally compact quantum groups
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Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• (λG,L2(G)) is the GNS representation for ϕ.
• U ∈ M(K(H)⊗ C0(G)): unitary corepn. of C0(G).(a) ξ ∈ H is s.t.b. a U-invariant if U(ξ ⊗ η) = ξ ⊗ η, ∀η ∈ L2(G).(b) A net {ξi}i∈I of unit vectors in H is called an almostU-invariant unit vector if ‖U(ξi ⊗ η)− ξi ⊗ η‖ → 0, ∀η ∈ L2(G).
• U has a non-zero invariant vector⇔ π1G ⊂ πU .
• U has almost invariant vectors⇔ π1G ≺ πU .
Ng Property T for locally compact quantum groups
![Page 27: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/27.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• (λG,L2(G)) is the GNS representation for ϕ.
• U ∈ M(K(H)⊗ C0(G)): unitary corepn. of C0(G).(a) ξ ∈ H is s.t.b. a U-invariant if U(ξ ⊗ η) = ξ ⊗ η, ∀η ∈ L2(G).(b) A net {ξi}i∈I of unit vectors in H is called an almostU-invariant unit vector if ‖U(ξi ⊗ η)− ξi ⊗ η‖ → 0, ∀η ∈ L2(G).
• U has a non-zero invariant vector⇔ π1G ⊂ πU .
• U has almost invariant vectors⇔ π1G ≺ πU .
Ng Property T for locally compact quantum groups
![Page 28: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/28.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• (λG,L2(G)) is the GNS representation for ϕ.
• U ∈ M(K(H)⊗ C0(G)): unitary corepn. of C0(G).(a) ξ ∈ H is s.t.b. a U-invariant if U(ξ ⊗ η) = ξ ⊗ η, ∀η ∈ L2(G).(b) A net {ξi}i∈I of unit vectors in H is called an almostU-invariant unit vector if ‖U(ξi ⊗ η)− ξi ⊗ η‖ → 0, ∀η ∈ L2(G).
• U has a non-zero invariant vector⇔ π1G ⊂ πU .
• U has almost invariant vectors⇔ π1G ≺ πU .
Ng Property T for locally compact quantum groups
![Page 29: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/29.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionG is said to have property T if every unit. corepn. of C0(G)having an almost invar. unit vector has a non-zero inv. vector.
The following extends some results in the paper by Kyed andSoltan:
PropositionThe following statements are equivalent.
(0) G has property T(1) Cu
0(G) ∼= kerπ1G ⊕ C.
(2) ∃ a proj. pG ∈ M(Cu0(G)) with pGCu
0(G)pG = CpG andπ1G(pG) = 1.
(3) π1G is an isolated point in Cu0(G).
Ng Property T for locally compact quantum groups
![Page 30: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/30.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionG is said to have property T if every unit. corepn. of C0(G)having an almost invar. unit vector has a non-zero inv. vector.
The following extends some results in the paper by Kyed andSoltan:
PropositionThe following statements are equivalent.
(0) G has property T(1) Cu
0(G) ∼= kerπ1G ⊕ C.
(2) ∃ a proj. pG ∈ M(Cu0(G)) with pGCu
0(G)pG = CpG andπ1G(pG) = 1.
(3) π1G is an isolated point in Cu0(G).
Ng Property T for locally compact quantum groups
![Page 31: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/31.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
For the corresp. of (T4) and (T5), we need the notion of “tensorproducts” and ”contragredients” of unit. corepns.
• If V is another unit. corepn. of C0(G) on a Hil. sp. K, thenU >©V := U13V23 ∈ M
(K(H⊗ K)⊗ C0(G)
)is a unit. corepn.
• Suppose G is of Kac type and κ : C0(G)→ C0(G) is theantipode. If τ : K(K)→ K(K) is the canon. anti-isom., thenV := (τ ⊗ κ)(V ) ∈ M(K(K)⊗ C0(G)) is a unit. corepn.• If one identifies H⊗ K with the space of Hilbert-Schmidtoperators from K to H, then
Lemma
Suppose that G is of Kac type. T ∈ H⊗ K is U >©V -invariant ifand only if U(T ⊗ 1)V ∗ = T ⊗ 1 (as operators from K⊗ L2(G) toH⊗ L2(G)).
Ng Property T for locally compact quantum groups
![Page 32: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/32.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
For the corresp. of (T4) and (T5), we need the notion of “tensorproducts” and ”contragredients” of unit. corepns.• If V is another unit. corepn. of C0(G) on a Hil. sp. K, thenU >©V := U13V23 ∈ M
(K(H⊗ K)⊗ C0(G)
)is a unit. corepn.
• Suppose G is of Kac type and κ : C0(G)→ C0(G) is theantipode. If τ : K(K)→ K(K) is the canon. anti-isom., thenV := (τ ⊗ κ)(V ) ∈ M(K(K)⊗ C0(G)) is a unit. corepn.• If one identifies H⊗ K with the space of Hilbert-Schmidtoperators from K to H, then
Lemma
Suppose that G is of Kac type. T ∈ H⊗ K is U >©V -invariant ifand only if U(T ⊗ 1)V ∗ = T ⊗ 1 (as operators from K⊗ L2(G) toH⊗ L2(G)).
Ng Property T for locally compact quantum groups
![Page 33: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/33.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
For the corresp. of (T4) and (T5), we need the notion of “tensorproducts” and ”contragredients” of unit. corepns.• If V is another unit. corepn. of C0(G) on a Hil. sp. K, thenU >©V := U13V23 ∈ M
(K(H⊗ K)⊗ C0(G)
)is a unit. corepn.
• Suppose G is of Kac type and κ : C0(G)→ C0(G) is theantipode. If τ : K(K)→ K(K) is the canon. anti-isom., thenV := (τ ⊗ κ)(V ) ∈ M(K(K)⊗ C0(G)) is a unit. corepn.
• If one identifies H⊗ K with the space of Hilbert-Schmidtoperators from K to H, then
Lemma
Suppose that G is of Kac type. T ∈ H⊗ K is U >©V -invariant ifand only if U(T ⊗ 1)V ∗ = T ⊗ 1 (as operators from K⊗ L2(G) toH⊗ L2(G)).
Ng Property T for locally compact quantum groups
![Page 34: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/34.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
For the corresp. of (T4) and (T5), we need the notion of “tensorproducts” and ”contragredients” of unit. corepns.• If V is another unit. corepn. of C0(G) on a Hil. sp. K, thenU >©V := U13V23 ∈ M
(K(H⊗ K)⊗ C0(G)
)is a unit. corepn.
• Suppose G is of Kac type and κ : C0(G)→ C0(G) is theantipode. If τ : K(K)→ K(K) is the canon. anti-isom., thenV := (τ ⊗ κ)(V ) ∈ M(K(K)⊗ C0(G)) is a unit. corepn.• If one identifies H⊗ K with the space of Hilbert-Schmidtoperators from K to H, then
Lemma
Suppose that G is of Kac type. T ∈ H⊗ K is U >©V -invariant ifand only if U(T ⊗ 1)V ∗ = T ⊗ 1 (as operators from K⊗ L2(G) toH⊗ L2(G)).
Ng Property T for locally compact quantum groups
![Page 35: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/35.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
For the corresp. of (T4) and (T5), we need the notion of “tensorproducts” and ”contragredients” of unit. corepns.• If V is another unit. corepn. of C0(G) on a Hil. sp. K, thenU >©V := U13V23 ∈ M
(K(H⊗ K)⊗ C0(G)
)is a unit. corepn.
• Suppose G is of Kac type and κ : C0(G)→ C0(G) is theantipode. If τ : K(K)→ K(K) is the canon. anti-isom., thenV := (τ ⊗ κ)(V ) ∈ M(K(K)⊗ C0(G)) is a unit. corepn.• If one identifies H⊗ K with the space of Hilbert-Schmidtoperators from K to H, then
Lemma
Suppose that G is of Kac type. T ∈ H⊗ K is U >©V -invariant ifand only if U(T ⊗ 1)V ∗ = T ⊗ 1 (as operators from K⊗ L2(G) toH⊗ L2(G)).
Ng Property T for locally compact quantum groups
![Page 36: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/36.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
G is of Kac type.
Propositionπ1G ⊂ πU >©V if and only if ∃ a fin. dim. unit. corepn. W s.t.πW ⊂ πU and πW ⊂ πV .
TheoremThen property T of G is also equivalent to the followingstatements.
(4) All fin. dimen. irred. repn. of Cu0(G) are isolated points in
Cu0(G).
(5) Cu0(G) ∼= B ⊕Mn(C) for a C∗-algebra B and n ∈ N.
Ng Property T for locally compact quantum groups
![Page 37: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/37.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
G is of Kac type.
Propositionπ1G ⊂ πU >©V if and only if ∃ a fin. dim. unit. corepn. W s.t.πW ⊂ πU and πW ⊂ πV .
TheoremThen property T of G is also equivalent to the followingstatements.
(4) All fin. dimen. irred. repn. of Cu0(G) are isolated points in
Cu0(G).
(5) Cu0(G) ∼= B ⊕Mn(C) for a C∗-algebra B and n ∈ N.
Ng Property T for locally compact quantum groups
![Page 38: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/38.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 39: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/39.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).
* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 40: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/40.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 41: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/41.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},
then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 42: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/42.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).
• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 43: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/43.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”
• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 44: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/44.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.
• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 45: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/45.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].
• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 46: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/46.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Let µ : G→ L(H) be a strongly cont. unit. repn. A continuousmap c : G→ H is called a
* 1-cocycle for µ if c(st) = µs(c(t)) + c(s) (s, t ∈ G).* 1-coboundary for µ if ∃ξ ∈ H s.t. c(s) = µs(ξ)− ξ (s ∈ G).
If Z 1(G;µ) := {1-cocycles} and B1(G;µ) := {1-coboundaries},then H1(G;µ) := Z 1(G;µ)/B1(G;µ) is called the firstcohomology of G with coefficient in (µ,H).• Problem: loc. comp. quant. groups “may not have point”• Suppose G is discrete and (Pol(G),∆,S, ε) is the Hopf ∗-alg.of matrix coefficents of G. Kyed defined a cohom. theory of(Pol(G), ε) with coeff. in ∗-repn of Pol(G), and showed thatproperty T of G is equiv. to the vanishing of all such cohom.• If G is a discrete group Γ, then Pol(G) = C[Γ].• How about locally compact groups?
Ng Property T for locally compact quantum groups
![Page 47: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/47.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 48: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/48.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 49: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/49.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 50: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/50.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 51: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/51.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 52: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/52.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;
2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 53: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/53.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.
3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for aunique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 54: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/54.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• G: loc. cpt. group with strongly cont. unit. repn.µ : G→ L(H).
• Cc(G) : all cont. funct. with compact support.
• µ : Cc(G)→ L(H) the ∗-repn induced by µ.
• εG : Cc(G)→ C canonical character.
• ∀ 1-cocycle c, define Φ(c)(f ) :=∫
G f (t)c(t)dt (f ∈ Cc(G))<–> “1-cocycle” (or “derivation”) for (µ, εG).
• Need a topology on Cc(G) satisfying1 any cont. ∗-repn of Cc(G) is of the form µ for a cont. unit.
repn of G;2 if c is a cont. 1-cocyc., then Φ(c) is cont.3 every cont. “1-cocyc. for (µ, εG)” is of the form Φ(c) for a
unique cont. 1-cocycle c for µ.
Ng Property T for locally compact quantum groups
![Page 55: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/55.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 56: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/56.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);
* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 57: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/57.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.
• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 58: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/58.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”
• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 59: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/59.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 60: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/60.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.
• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 61: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/61.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.
• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 62: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/62.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).
(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0,∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 63: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/63.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• B is a ∗-alg., ε : B → C a ∗-char. and π : B → L(H) a ∗-repn.
• θ : B → H is called a* 1-cocycle for (π, ε) if θ(ab) = π(a)θ(b) + θ(a)ε(b) (a,b ∈ B);* 1-coboundary for (π, ε) if ∃ξ ∈ H with θ(a) = ξε(a)− µ(a)ξ.• 1-cocycle for (π, ε) = “(π, ε)-derivation”• 1-coboundary for (π, ε) = “inner (π, ε)-derivation”
• K(G) := {non-empty compact subsets of G}.• ∀K ∈ K(G), set CK (G) := {cont. funct. with support in K}.• A linear map θ from Cc(G) to a normed space E is s.t.b:(a) locally bounded, if θ|CK (G) is L1-bounded ∀K ∈ K(G).(b) continuously locally bounded, if ∃ a family {κK}K∈K(G) in R+:
* ‖θ(f )‖H ≤ κK‖f‖L1(G) (K ∈ K(G); f ∈ CK (G)),* ∀ε > 0, ∃ a compact neigh. U of e s.t. κK < ε if K ⊆ U.
Ng Property T for locally compact quantum groups
![Page 64: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/64.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Proposition
If π : Cc(G)→ L(H) is a non-deg. ∗-repn., ∃ a stronly cont. unit.repn. µ of G on H with π = µ if and only if π is loc. bdd.
Theoremθ : Cc(G)→ H a loc. bdd. 1-cocyc. for (µ, εG);c : G→ H a cont. 1-cocyc. for µ.(a) ∃ weakly cont. 1-cocyc. Ψ(θ) for µ s.t.〈θ(g), ζ〉 =
∫G g(t)〈Ψ(θ)(t), ζ〉dt (g ∈ Cc(G); ζ ∈ H).
(b) Φ(c) is cont. loc. bdd. 1-cocyc. for (µ, εG) s.t. Ψ(Φ(c)) = c.(c) If θ is cont. loc. bdd., then Ψ(θ) is cont. and Φ(Ψ(θ)) = θ.(d) θ is L1-bounded if and only if Ψ(θ) is a 1-coboundary.
Ng Property T for locally compact quantum groups
![Page 65: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/65.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Proposition
If π : Cc(G)→ L(H) is a non-deg. ∗-repn., ∃ a stronly cont. unit.repn. µ of G on H with π = µ if and only if π is loc. bdd.
Theoremθ : Cc(G)→ H a loc. bdd. 1-cocyc. for (µ, εG);c : G→ H a cont. 1-cocyc. for µ.
(a) ∃ weakly cont. 1-cocyc. Ψ(θ) for µ s.t.〈θ(g), ζ〉 =
∫G g(t)〈Ψ(θ)(t), ζ〉dt (g ∈ Cc(G); ζ ∈ H).
(b) Φ(c) is cont. loc. bdd. 1-cocyc. for (µ, εG) s.t. Ψ(Φ(c)) = c.(c) If θ is cont. loc. bdd., then Ψ(θ) is cont. and Φ(Ψ(θ)) = θ.(d) θ is L1-bounded if and only if Ψ(θ) is a 1-coboundary.
Ng Property T for locally compact quantum groups
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Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Proposition
If π : Cc(G)→ L(H) is a non-deg. ∗-repn., ∃ a stronly cont. unit.repn. µ of G on H with π = µ if and only if π is loc. bdd.
Theoremθ : Cc(G)→ H a loc. bdd. 1-cocyc. for (µ, εG);c : G→ H a cont. 1-cocyc. for µ.(a) ∃ weakly cont. 1-cocyc. Ψ(θ) for µ s.t.〈θ(g), ζ〉 =
∫G g(t)〈Ψ(θ)(t), ζ〉dt (g ∈ Cc(G); ζ ∈ H).
(b) Φ(c) is cont. loc. bdd. 1-cocyc. for (µ, εG) s.t. Ψ(Φ(c)) = c.(c) If θ is cont. loc. bdd., then Ψ(θ) is cont. and Φ(Ψ(θ)) = θ.(d) θ is L1-bounded if and only if Ψ(θ) is a 1-coboundary.
Ng Property T for locally compact quantum groups
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Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Proposition
If π : Cc(G)→ L(H) is a non-deg. ∗-repn., ∃ a stronly cont. unit.repn. µ of G on H with π = µ if and only if π is loc. bdd.
Theoremθ : Cc(G)→ H a loc. bdd. 1-cocyc. for (µ, εG);c : G→ H a cont. 1-cocyc. for µ.(a) ∃ weakly cont. 1-cocyc. Ψ(θ) for µ s.t.〈θ(g), ζ〉 =
∫G g(t)〈Ψ(θ)(t), ζ〉dt (g ∈ Cc(G); ζ ∈ H).
(b) Φ(c) is cont. loc. bdd. 1-cocyc. for (µ, εG) s.t. Ψ(Φ(c)) = c.
(c) If θ is cont. loc. bdd., then Ψ(θ) is cont. and Φ(Ψ(θ)) = θ.(d) θ is L1-bounded if and only if Ψ(θ) is a 1-coboundary.
Ng Property T for locally compact quantum groups
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Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Proposition
If π : Cc(G)→ L(H) is a non-deg. ∗-repn., ∃ a stronly cont. unit.repn. µ of G on H with π = µ if and only if π is loc. bdd.
Theoremθ : Cc(G)→ H a loc. bdd. 1-cocyc. for (µ, εG);c : G→ H a cont. 1-cocyc. for µ.(a) ∃ weakly cont. 1-cocyc. Ψ(θ) for µ s.t.〈θ(g), ζ〉 =
∫G g(t)〈Ψ(θ)(t), ζ〉dt (g ∈ Cc(G); ζ ∈ H).
(b) Φ(c) is cont. loc. bdd. 1-cocyc. for (µ, εG) s.t. Ψ(Φ(c)) = c.(c) If θ is cont. loc. bdd., then Ψ(θ) is cont. and Φ(Ψ(θ)) = θ.
(d) θ is L1-bounded if and only if Ψ(θ) is a 1-coboundary.
Ng Property T for locally compact quantum groups
![Page 69: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/69.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Proposition
If π : Cc(G)→ L(H) is a non-deg. ∗-repn., ∃ a stronly cont. unit.repn. µ of G on H with π = µ if and only if π is loc. bdd.
Theoremθ : Cc(G)→ H a loc. bdd. 1-cocyc. for (µ, εG);c : G→ H a cont. 1-cocyc. for µ.(a) ∃ weakly cont. 1-cocyc. Ψ(θ) for µ s.t.〈θ(g), ζ〉 =
∫G g(t)〈Ψ(θ)(t), ζ〉dt (g ∈ Cc(G); ζ ∈ H).
(b) Φ(c) is cont. loc. bdd. 1-cocyc. for (µ, εG) s.t. Ψ(Φ(c)) = c.(c) If θ is cont. loc. bdd., then Ψ(θ) is cont. and Φ(Ψ(θ)) = θ.(d) θ is L1-bounded if and only if Ψ(θ) is a 1-coboundary.
Ng Property T for locally compact quantum groups
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Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionBP : loc. convex ∗-alg.; ε : B → C a P-cont. ∗-character. We saythat (BP, ε) has property (FH) if ∀P-cont. non-deg. ∗-repnπ : B → L(H), any P-cont. 1-cocycle for (π, ε) is a1-coboundary.
• For Ban. alg., prop. (FH) is a weaker version of ε-amenability.• If B is a unital Ban. ∗-alg. and (B‖·‖, ε) has prop. (FH), thenker ε = ker ε · ker ε.• If B is a Ban. ∗-alg. and ker ε has a bounded approximateidentity (e.g. when B is a C∗-alg), then (B‖·‖, ε) has prop. (FH).• If B is a left amen. F -alg. with isom. invol. and ε ∈ B∗ is theidentity with ε(b∗) = ε(b)∗, then (B‖·‖, ε) has prop. (FH).• If G is an amen. loc. comp. quant. gp. of Kac type, then(L1(G)‖·‖L1(G)
, π1G) has prop. (FH).
Ng Property T for locally compact quantum groups
![Page 71: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/71.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionBP : loc. convex ∗-alg.; ε : B → C a P-cont. ∗-character. We saythat (BP, ε) has property (FH) if ∀P-cont. non-deg. ∗-repnπ : B → L(H), any P-cont. 1-cocycle for (π, ε) is a1-coboundary.
• For Ban. alg., prop. (FH) is a weaker version of ε-amenability.
• If B is a unital Ban. ∗-alg. and (B‖·‖, ε) has prop. (FH), thenker ε = ker ε · ker ε.• If B is a Ban. ∗-alg. and ker ε has a bounded approximateidentity (e.g. when B is a C∗-alg), then (B‖·‖, ε) has prop. (FH).• If B is a left amen. F -alg. with isom. invol. and ε ∈ B∗ is theidentity with ε(b∗) = ε(b)∗, then (B‖·‖, ε) has prop. (FH).• If G is an amen. loc. comp. quant. gp. of Kac type, then(L1(G)‖·‖L1(G)
, π1G) has prop. (FH).
Ng Property T for locally compact quantum groups
![Page 72: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/72.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionBP : loc. convex ∗-alg.; ε : B → C a P-cont. ∗-character. We saythat (BP, ε) has property (FH) if ∀P-cont. non-deg. ∗-repnπ : B → L(H), any P-cont. 1-cocycle for (π, ε) is a1-coboundary.
• For Ban. alg., prop. (FH) is a weaker version of ε-amenability.• If B is a unital Ban. ∗-alg. and (B‖·‖, ε) has prop. (FH), thenker ε = ker ε · ker ε.
• If B is a Ban. ∗-alg. and ker ε has a bounded approximateidentity (e.g. when B is a C∗-alg), then (B‖·‖, ε) has prop. (FH).• If B is a left amen. F -alg. with isom. invol. and ε ∈ B∗ is theidentity with ε(b∗) = ε(b)∗, then (B‖·‖, ε) has prop. (FH).• If G is an amen. loc. comp. quant. gp. of Kac type, then(L1(G)‖·‖L1(G)
, π1G) has prop. (FH).
Ng Property T for locally compact quantum groups
![Page 73: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/73.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionBP : loc. convex ∗-alg.; ε : B → C a P-cont. ∗-character. We saythat (BP, ε) has property (FH) if ∀P-cont. non-deg. ∗-repnπ : B → L(H), any P-cont. 1-cocycle for (π, ε) is a1-coboundary.
• For Ban. alg., prop. (FH) is a weaker version of ε-amenability.• If B is a unital Ban. ∗-alg. and (B‖·‖, ε) has prop. (FH), thenker ε = ker ε · ker ε.• If B is a Ban. ∗-alg. and ker ε has a bounded approximateidentity (e.g. when B is a C∗-alg), then (B‖·‖, ε) has prop. (FH).
• If B is a left amen. F -alg. with isom. invol. and ε ∈ B∗ is theidentity with ε(b∗) = ε(b)∗, then (B‖·‖, ε) has prop. (FH).• If G is an amen. loc. comp. quant. gp. of Kac type, then(L1(G)‖·‖L1(G)
, π1G) has prop. (FH).
Ng Property T for locally compact quantum groups
![Page 74: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/74.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionBP : loc. convex ∗-alg.; ε : B → C a P-cont. ∗-character. We saythat (BP, ε) has property (FH) if ∀P-cont. non-deg. ∗-repnπ : B → L(H), any P-cont. 1-cocycle for (π, ε) is a1-coboundary.
• For Ban. alg., prop. (FH) is a weaker version of ε-amenability.• If B is a unital Ban. ∗-alg. and (B‖·‖, ε) has prop. (FH), thenker ε = ker ε · ker ε.• If B is a Ban. ∗-alg. and ker ε has a bounded approximateidentity (e.g. when B is a C∗-alg), then (B‖·‖, ε) has prop. (FH).• If B is a left amen. F -alg. with isom. invol. and ε ∈ B∗ is theidentity with ε(b∗) = ε(b)∗, then (B‖·‖, ε) has prop. (FH).
• If G is an amen. loc. comp. quant. gp. of Kac type, then(L1(G)‖·‖L1(G)
, π1G) has prop. (FH).
Ng Property T for locally compact quantum groups
![Page 75: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/75.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
DefinitionBP : loc. convex ∗-alg.; ε : B → C a P-cont. ∗-character. We saythat (BP, ε) has property (FH) if ∀P-cont. non-deg. ∗-repnπ : B → L(H), any P-cont. 1-cocycle for (π, ε) is a1-coboundary.
• For Ban. alg., prop. (FH) is a weaker version of ε-amenability.• If B is a unital Ban. ∗-alg. and (B‖·‖, ε) has prop. (FH), thenker ε = ker ε · ker ε.• If B is a Ban. ∗-alg. and ker ε has a bounded approximateidentity (e.g. when B is a C∗-alg), then (B‖·‖, ε) has prop. (FH).• If B is a left amen. F -alg. with isom. invol. and ε ∈ B∗ is theidentity with ε(b∗) = ε(b)∗, then (B‖·‖, ε) has prop. (FH).• If G is an amen. loc. comp. quant. gp. of Kac type, then(L1(G)‖·‖L1(G)
, π1G) has prop. (FH).
Ng Property T for locally compact quantum groups
![Page 76: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/76.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Consider T to be the loc. convex inductive top. on Cc(G)induced by the sys.
{(CK (G), ‖ · ‖L1(G))
}K∈K(G)
of normed sp.
• A linear map Ψ from Cc(G) to a normed sp E is loc. bdd⇔ itis T-cont.
Theorem
Let G be a loc. comp. group.(a) If G is 2nd countable, then (Cc(G)T, εG) has prop. (FH)⇔ G has prop. T .(b) (L1(G)‖·‖L1(G)
, εG) always has prop. (FH).
(c) (A(G)‖·‖, δe) has prop. (FH).
• One may also define char. amen. for loc. conv. ∗-alg. If G is2nd count., Cc(G)T is εG-amen. if and only if G is compact.
Ng Property T for locally compact quantum groups
![Page 77: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/77.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Consider T to be the loc. convex inductive top. on Cc(G)induced by the sys.
{(CK (G), ‖ · ‖L1(G))
}K∈K(G)
of normed sp.
• A linear map Ψ from Cc(G) to a normed sp E is loc. bdd⇔ itis T-cont.
Theorem
Let G be a loc. comp. group.(a) If G is 2nd countable, then (Cc(G)T, εG) has prop. (FH)⇔ G has prop. T .(b) (L1(G)‖·‖L1(G)
, εG) always has prop. (FH).
(c) (A(G)‖·‖, δe) has prop. (FH).
• One may also define char. amen. for loc. conv. ∗-alg. If G is2nd count., Cc(G)T is εG-amen. if and only if G is compact.
Ng Property T for locally compact quantum groups
![Page 78: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/78.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Consider T to be the loc. convex inductive top. on Cc(G)induced by the sys.
{(CK (G), ‖ · ‖L1(G))
}K∈K(G)
of normed sp.
• A linear map Ψ from Cc(G) to a normed sp E is loc. bdd⇔ itis T-cont.
Theorem
Let G be a loc. comp. group.(a) If G is 2nd countable, then (Cc(G)T, εG) has prop. (FH)⇔ G has prop. T .
(b) (L1(G)‖·‖L1(G), εG) always has prop. (FH).
(c) (A(G)‖·‖, δe) has prop. (FH).
• One may also define char. amen. for loc. conv. ∗-alg. If G is2nd count., Cc(G)T is εG-amen. if and only if G is compact.
Ng Property T for locally compact quantum groups
![Page 79: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/79.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Consider T to be the loc. convex inductive top. on Cc(G)induced by the sys.
{(CK (G), ‖ · ‖L1(G))
}K∈K(G)
of normed sp.
• A linear map Ψ from Cc(G) to a normed sp E is loc. bdd⇔ itis T-cont.
Theorem
Let G be a loc. comp. group.(a) If G is 2nd countable, then (Cc(G)T, εG) has prop. (FH)⇔ G has prop. T .(b) (L1(G)‖·‖L1(G)
, εG) always has prop. (FH).
(c) (A(G)‖·‖, δe) has prop. (FH).
• One may also define char. amen. for loc. conv. ∗-alg. If G is2nd count., Cc(G)T is εG-amen. if and only if G is compact.
Ng Property T for locally compact quantum groups
![Page 80: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/80.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Consider T to be the loc. convex inductive top. on Cc(G)induced by the sys.
{(CK (G), ‖ · ‖L1(G))
}K∈K(G)
of normed sp.
• A linear map Ψ from Cc(G) to a normed sp E is loc. bdd⇔ itis T-cont.
Theorem
Let G be a loc. comp. group.(a) If G is 2nd countable, then (Cc(G)T, εG) has prop. (FH)⇔ G has prop. T .(b) (L1(G)‖·‖L1(G)
, εG) always has prop. (FH).
(c) (A(G)‖·‖, δe) has prop. (FH).
• One may also define char. amen. for loc. conv. ∗-alg. If G is2nd count., Cc(G)T is εG-amen. if and only if G is compact.
Ng Property T for locally compact quantum groups
![Page 81: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/81.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
• Consider T to be the loc. convex inductive top. on Cc(G)induced by the sys.
{(CK (G), ‖ · ‖L1(G))
}K∈K(G)
of normed sp.
• A linear map Ψ from Cc(G) to a normed sp E is loc. bdd⇔ itis T-cont.
Theorem
Let G be a loc. comp. group.(a) If G is 2nd countable, then (Cc(G)T, εG) has prop. (FH)⇔ G has prop. T .(b) (L1(G)‖·‖L1(G)
, εG) always has prop. (FH).
(c) (A(G)‖·‖, δe) has prop. (FH).
• One may also define char. amen. for loc. conv. ∗-alg. If G is2nd count., Cc(G)T is εG-amen. if and only if G is compact.
Ng Property T for locally compact quantum groups
![Page 82: Property T for locally compact quantum groups · Property T for locally compact quantum groups Chi-Keung Ng The Chern Institute of Mathematics Nankai University Workshop on Operator](https://reader034.vdocument.in/reader034/viewer/2022042312/5eda7aafb3745412b57169b9/html5/thumbnails/82.jpg)
Background and preliminaryProperty T and some equivalent forms
Delorme-Guichardet type theorem?
Reference:1. X. Chen and C.K. Ng, Property T for general locally compactquantum groups, preprint.
2. X. Chen, A.T.M. Lau and C.K. Ng, Convolution algebras andproperty (T) for locally compact groups, in preparation.
Thanks
Ng Property T for locally compact quantum groups