Topological centres for group algebras, actions, and quantum groups

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Topological centres for group algebras, actions, and quantum groups

Matthias Neufang Carleton University (Ottawa)

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Topological centre basics Topological centre problems Topological centres as a tool

1 Topological centre basics

2 Topological centre problems

3 Topological centres as a tool

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Arens products: Algebraic description

ABanach algebra; as Banach space: A,→ A^∗∗

∃2 canonical extensions of product toA^∗∗ (Arens’51) X,Y ∈ A^∗∗,f ∈ A^∗,a,b ∈ A

hX2Y,fi = hX,Y2fi hY2f,ai = hY,f2ai

hf2a,bi = hf,a·bi . . . and the other way around:

hX3Y,fi = hY,f3Xi hf3X,ai = hX,a3fi

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

∃2 canonical extensions of product toA^∗∗ (Arens’51)

X,Y ∈ A^∗∗,f ∈ A^∗,a,b ∈ A

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

hX2Y,fi = hX,Y2fi

hY2f,ai = hY,f2ai hf2a,bi = hf,a·bi . . . and the other way around:

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

hf2a,bi = hf,a·bi

. . . and the other way around:

ha3f,bi = hf,b·ai

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Arens products: Algebraic description

ha3f,bi = hf,b·ai

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Arens products: Topological description

A 3x_i −→ X ∈ A^∗∗ (w^∗) A 3yj −→ Y ∈ A^∗∗ (w^∗) X2Y = lim_i lim_j x_i·y_j X3Y = lim_j lim_i x_i ·y_j

2=3 ⇔: AArens regular (e.g., operator algebras) But for algebras closest to the heart of harmonic analysts:

X2Y 6= X3Y

;How to measure the degree of non-regularity?

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Arens products: Topological description

A 3x_i −→ X ∈ A^∗∗ (w^∗) A 3yj −→ Y ∈ A^∗∗ (w^∗)

X2Y = lim_i lim_j x_i·y_j X3Y = lim_j lim_i x_i ·y_j

X2Y 6= X3Y

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Arens products: Topological description

A 3x_i −→ X ∈ A^∗∗ (w^∗) A 3yj −→ Y ∈ A^∗∗ (w^∗) X2Y = lim_i lim_j x_i ·y_j X3Y = lim_j lim_i x_i ·y_j

X2Y 6= X3Y

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Arens products: Topological description

2=3 ⇔: AArens regular (e.g., operator algebras)

But for algebras closest to the heart of harmonic analysts: X2Y 6= X3Y

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Arens products: Topological description

X2Y 6= X3Y

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Arens products: Topological description

X2Y 6= X3Y

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

= { X | Y 7→X2Y w^∗-cont.} Zr(A^∗∗) := { X | Y2X = Y3X ∀ Y }

= { X | Y 7→Y3X w^∗-cont.} AArens regular :⇔ Z` =Zr =A^∗∗

Definition (Dales–Lau ’05)

ALeft StronglyIrregular (LSI) :⇔ Z`=A ARight StronglyIrregular (RSI) :⇔ Zr =A AStronglyIrregular (SI) :⇔ Z_` =Z_r =A

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

= { X | Y 7→X2Y w^∗-cont.} Zr(A^∗∗) := { X | Y2X = Y3X ∀ Y }

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

= { X | Y 7→X2Y w^∗-cont.}

Zr(A^∗∗) := { X | Y2X = Y3X ∀ Y }

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

= { X | Y 7→X2Y w^∗-cont.} Z_r(A^∗∗) := { X | Y2X = Y3X ∀ Y }

= { X | Y 7→Y3X w^∗-cont.}

AArens regular :⇔ Z` =Zr =A^∗∗ Definition (Dales–Lau ’05)

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

ALeft StronglyIrregular (LSI) :⇔ Z`=A

ARight StronglyIrregular (RSI) :⇔ Zr =A AStronglyIrregular (SI) :⇔ Z_` =Z_r =A

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

ALeft StronglyIrregular (LSI) :⇔ Z`=A ARight StronglyIrregular (RSI) :⇔ Zr =A

AStronglyIrregular (SI) :⇔ Z_` =Z_r =A

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Topological centres

Z_`(A^∗∗) := { X | X2Y = X3Y ∀ Y }

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Asymmetries

Obviously: Z_`=A^∗∗⇔ Z_r =A^∗∗ Question (Dales–Lau ’05)

LSI⇒RSI ? Proposition (N. ’05)

No!

Example: convolution algebraT(G) = (T(L2(G)),∗) ρ∗τ :=

Z

G

L_xρL_x⁻¹ π(τ)(x)dx

where π:T(G)→L₁(G) canonical quotient map

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Asymmetries

Obviously: Z_`=A^∗∗ ⇔ Z_r =A^∗∗

Question (Dales–Lau ’05)

No!

Z

G

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Asymmetries

LSI⇒RSI ?

Proposition (N. ’05)

No!

Z

G

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Asymmetries

No!

Z

G

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Arens products & Kadison–Singer Conjecture

Kadison–Singer Conjecture (’59)

Letm∈`∞(N)^∗ be a pure state, i.e.,m∈βN. Doesm extend uniquely to a state on B(`₂(N)) ?

Equivalent conjecture about Bessel sequences (Weaver ’04)

∃M ≥2 andε >0 such that:

givenx₁, . . . ,x_n∈C^k (n≥2) with `₂-norm ≤1 and X

i

|hx_i,yi|²≤M ∀ unit vectory ∈C^k

⇒ ∃partitionA₁, . . . ,A_` (`≥2) of{1, . . . ,n} with X

i∈A_j

|hx_i,yi|²≤M−ε ∀ unit vectorsy ∈C^k, ∀ j

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Arens products & Kadison–Singer Conjecture

Letm∈`∞(N)^∗ be a pure state, i.e.,m∈βN.

Doesm extend uniquely to a state on B(`₂(N)) ?

i

i∈A_j

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Arens products & Kadison–Singer Conjecture

i

i∈A_j

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Arens products & Kadison–Singer Conjecture

i

i∈A_j

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Arens products & Kadison–Singer Conjecture

i

i∈A_j

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Arens products & Kadison–Singer Conjecture

givenx₁, . . . ,x_n∈C^k (n≥2) with `₂-norm ≤1

and X

i

i∈A_j

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Arens products & Kadison–Singer Conjecture

i

|hx_i,yi|² ≤M ∀ unit vectory ∈C^k

i∈A_j

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Arens products & Kadison–Singer Conjecture

i

⇒ ∃partitionA₁, . . . ,A_` (`≥2) of{1, . . . ,n}

with X

i∈A_j

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Arens products & Kadison–Singer Conjecture

i

⇒ ∃partitionA₁, . . . ,A_` (`≥2) of{1, . . . ,n}with X

i∈A_j

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Kadison–Singer Conjecture & T (G)

Kadison–Singer Conjecture for discreteG

⇒the map

βG 3m 7→ me ∈ T(G)^∗∗ ismultiplicative w.r.t. convolution

This is true forG=Z as well asG=F2 . . .

;Transfer of topological dynamics!

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Kadison–Singer Conjecture & T (G)

⇒the map

βG 3m 7→ me ∈ T(G)^∗∗ ismultiplicative w.r.t. convolution

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Kadison–Singer Conjecture & T (G)

⇒the map

βG 3m 7→ me ∈ T(G)^∗∗

ismultiplicative w.r.t. convolution

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Kadison–Singer Conjecture & T (G)

⇒the map

βG 3m 7→ me ∈ T(G)^∗∗

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Kadison–Singer Conjecture & T (G)

⇒the map

βG 3m 7→ me ∈ T(G)^∗∗

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Kadison–Singer Conjecture & T (G)

⇒the map

βG 3m 7→ me ∈ T(G)^∗∗

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The Ghahramani–Lau Conjecture

Theorem (Lau–Losert ’88)

L₁(G)is SI for any locally compact groupG. Conjecture (Lau ’94 & Ghahramani–Lau ’95) M(G) is SI for any locally compact group G. Theorem (Losert–N.–Pachl–Stepr¯ans ’09) The conjecture holds in full generality.

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The Ghahramani–Lau Conjecture

L₁(G)is SI for any locally compact groupG.

Conjecture (Lau ’94 & Ghahramani–Lau ’95) M(G) is SI for any locally compact group G. Theorem (Losert–N.–Pachl–Stepr¯ans ’09) The conjecture holds in full generality.

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The Ghahramani–Lau Conjecture

Conjecture (Lau ’94 & Ghahramani–Lau ’95) M(G) is SI for any locally compact group G.

Theorem (Losert–N.–Pachl–Stepr¯ans ’09) The conjecture holds in full generality.

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The Ghahramani–Lau Conjecture

Conjecture (Lau ’94 & Ghahramani–Lau ’95) M(G) is SI for any locally compact group G.

Theorem (Losert–N.–Pachl–Stepr¯ans ’09) The conjecture holds in full generality.

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Key technique: Factorization

Definition (N. ’05) ABanach algebra, κ≥ ℵ₀.

1 A hasfactorization propertyof levelκ (Fκ) if

∀ (h_i)i∈I ⊆Ball(A^∗), |I| ≤κ

∃ (X_i)_i∈I ⊆Ball(A^∗∗) ∃h∈ A^∗

h_i =X_i2h (i ∈I)

2 A hasMazur’s propertyof levelκ (M_κ) if

any X ∈ A^∗∗ which isw^∗-κ-continuous on A^∗, lies inA. Theorem (N. ’05)

Ahas F_κ and M_κ for someκ≥ ℵ₀ ⇒ Ais SI

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Key technique: Factorization

∀ (h_i)i∈I ⊆Ball(A^∗), |I| ≤κ

∃ (X_i)_i∈I ⊆Ball(A^∗∗) ∃h∈ A^∗

h_i =X_i2h (i ∈I)

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Key technique: Factorization

∀ (h_i)i∈I ⊆Ball(A^∗), |I| ≤κ

∃ (X_i)_i∈I ⊆Ball(A^∗∗) ∃h∈ A^∗

h_i =X_i2h (i ∈I)

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Key technique: Factorization

∀ (h_i)i∈I ⊆Ball(A^∗), |I| ≤κ

∃ (X_i)_i∈I ⊆Ball(A^∗∗) ∃ h∈ A^∗

h_i =X_i2h (i ∈I)

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Key technique: Factorization

∀ (h_i)i∈I ⊆Ball(A^∗), |I| ≤κ

∃ (X_i)_i∈I ⊆Ball(A^∗∗) ∃ h∈ A^∗

h_i =X_i2h (i ∈I)

any X ∈ A^∗∗ which isw^∗-κ-continuous onA^∗, lies inA.

Theorem (N. ’05)

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Key technique: Factorization

∀ (h_i)i∈I ⊆Ball(A^∗), |I| ≤κ

∃ (X_i)_i∈I ⊆Ball(A^∗∗) ∃ h∈ A^∗

h_i =X_i2h (i ∈I)

any X ∈ A^∗∗ which isw^∗-κ-continuous onA^∗, lies inA.

Theorem (N. ’05)

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The dual setting

Problem (Cechini–Zappa ’81)

Consider the Fourier algebraA(G) ={hL_(·)ξ, ηi |ξ, η∈L₂(G)}. IsA(G) SI?

Yesfor large classes of amenable groups. Theorem (Losert ’02 & ’04)

Nofor G=F2 and also forG =SU(3)! Theorem (Filali–Monfared–N. ’10)

Yesfor SU(3)^ℵ¹ and SU(3)^c – in fact, for any compact group s.t. b(G)has uncountable cofinality.

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The dual setting

Consider the Fourier algebraA(G) ={hL_(·)ξ, ηi |ξ, η∈L₂(G)}.

IsA(G) SI?

Yesfor large classes of amenable groups. Theorem (Losert ’02 & ’04)

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The dual setting

IsA(G) SI?

Yesfor large classes of amenable groups.

Theorem (Losert ’02 & ’04)

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The dual setting

IsA(G) SI?

Nofor G=F2 and also forG =SU(3) !

Theorem (Filali–Monfared–N. ’10)

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The dual setting

IsA(G) SI?

Nofor G=F2 and also forG =SU(3) ! Theorem (Filali–Monfared–N. ’10)

Yesfor SU(3)^ℵ¹ and SU(3)^c – in fact, for any compact group s.t.

b(G)has uncountable cofinality.

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

G compact s.t. b(G) has uncountable cofinality. Then:

∀(Tα)α∈I ⊆BallL(G) with |I| ≤b(G)

∃(X_α)_α∈I ⊆BallL(G)^∗

∃T ∈ L(G) s.t.

T_α = X_α 2T So A(G) has F_b(G).

Theorem (Hu–N. ’06) A(G)has M_b(G).

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

G compact s.t. b(G)has uncountable cofinality.

Then:

∃T ∈ L(G) s.t.

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

G compact s.t. b(G)has uncountable cofinality. Then:

∃T ∈ L(G) s.t.

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

∃(Xα)α∈I ⊆BallL(G)^∗

∃T ∈ L(G) s.t.

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

∃T ∈ L(G) s.t.

T_α = X_α 2T

So A(G) has F_b(G). Theorem (Hu–N. ’06) A(G)has M_b(G).

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

∃T ∈ L(G) s.t.

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Proof:

Factorization & Mazur’s property for A(G) = L(G)

_∗

∃T ∈ L(G) s.t.

Theorem (Hu–N. ’06) A(G) has M_b(G).

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Our strategy to get operator factorization

Tα = Xα 2T

use central projections in L(G) arising from family of compact normal subgroups of G decreasing toe, of sizeI =b(G) double the index set I: α ; (α,β)

“separate” projections by coordinate functions of a b(G)-family of irred. unitary rep’s of G

this gives a family of pairwise orthogonal projections in L(G) truncateTα with these and sum up (w^∗) ;operator T X_α arew^∗-cluster points in L(G)^∗ of rep’s alongβ

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Our strategy to get operator factorization

Tα = Xα 2T

use central projections in L(G) arising from family of compact normal subgroups of G decreasing toe, of sizeI =b(G)

double the index set I: α ; (α,β)

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Our strategy to get operator factorization

Tα = Xα 2T

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Our strategy to get operator factorization

Tα = Xα 2T

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Our strategy to get operator factorization

Tα = Xα 2T

this gives a family of pairwise orthogonal projections in L(G)

truncateTα with these and sum up (w^∗) ;operator T X_α arew^∗-cluster points in L(G)^∗ of rep’s alongβ

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Our strategy to get operator factorization

Tα = Xα 2T

this gives a family of pairwise orthogonal projections in L(G) truncateTα with these and sum up (w^∗); operator T

X_α arew^∗-cluster points in L(G)^∗ of rep’s alongβ

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Our strategy to get operator factorization

Tα = Xα 2T

this gives a family of pairwise orthogonal projections in L(G) truncateTα with these and sum up (w^∗); operator T Xα arew^∗-cluster points in L(G)^∗ of rep’s alongβ

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Topological centres and multipliers

Problem (Lau– ¨Ulger ’96)

ABanach algebra with BAI s.t. A^∗ vN algebra. Let X ∈Zr(A^∗∗) . ConsiderX₂:A^∗ 3h7→X2h∈ A^∗ .

AreKer(X₂) andX₂(BallA^∗) w^∗-closed? Theorem (Hu–N.–Ruan ’09)

Nofor A=A(SU(3))

Proof: CombineLosert’s resultZ(A^∗∗)6=Awith the following Theorem (Hu–N.–Ruan ’09)

AssumeG metrizable andσ-compact. For X ∈Z(A(G)^∗∗): X ∈A(G) ⇔ Ker(X₂) and X₂(BallL(G))are w^∗-closed This relies on work byGodefroy–Talagrand ’89 & N. ’01

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Topological centres and multipliers

ABanach algebra with BAI s.t. A^∗ vN algebra. Let X ∈Zr(A^∗∗) . ConsiderX₂ :A^∗ 3h7→X2h∈ A^∗ .

AreKer(X₂) andX₂(BallA^∗) w^∗-closed? Theorem (Hu–N.–Ruan ’09)

Nofor A=A(SU(3))

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Topological centres and multipliers

AreKer(X₂) andX₂(BallA^∗) w^∗-closed?

Theorem (Hu–N.–Ruan ’09) Nofor A=A(SU(3))

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Topological centres and multipliers

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Topological centres and multipliers

Proof: CombineLosert’s result Z(A^∗∗)6=Awith the following Theorem (Hu–N.–Ruan ’09)

AssumeG metrizable andσ-compact. For X ∈Z(A(G)^∗∗):

X ∈A(G) ⇔ Ker(X₂) and X₂(BallL(G))are w^∗-closed This relies on work byGodefroy–Talagrand ’89 & N. ’01

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Ideals and multipliers

Theorem (Mbekhta–N. ’10)

A ideal in its bidual (OK for A=L₁(G) iff G is compact) A has BRAI

⇒ ∀I left ideal with BRAI

∃ idempotent p∈ RM(A) s.t. I =p(A)

Thisgeneralizesresults of Reiter (’71) and Bekka (’90) onA=L1(G) for compactG.

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Ideals and multipliers

A ideal in its bidual (OK forA=L₁(G) iff G is compact)

A has BRAI

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Ideals and multipliers

A ideal in its bidual (OK forA=L₁(G) iff G is compact) A has BRAI

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Ideals and multipliers

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Ideals and multipliers

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Ideals and multipliers

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Group actions and invariant means

Solution to the Banach–Ruziewicz Problem (Banach ’23; Margulis/Sullivan ’80/’81; Drinfeld ’84)

Except for n= 1, Lebesgue measure is the onlyinvariant mean on L∞(Sⁿ) for the action of O(n+ 1).

What about thediscretesituation?

Of course, forGyG: ∃! inv. meanon`∞(G) ⇔ G finite Quick proof using topological centres (Lau ’86):

uniqueinv. meanM

⇒ M ∈Z_`(`∞(G)^∗) =`₁(G)

⇒ M finite Haar measure, soG finite! What about general actionsGyX ?

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Group actions and invariant means

Solution to the Banach–Ruziewicz Problem (Banach ’23;

Margulis/Sullivan ’80/’81; Drinfeld ’84)

uniqueinv. meanM

⇒ M ∈Z_`(`∞(G)^∗) =`₁(G)

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Group actions and invariant means

uniqueinv. meanM

⇒ M ∈Z_`(`∞(G)^∗) =`₁(G)

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Group actions and invariant means

Of course, forGyG: ∃! inv. meanon`∞(G)⇔ G finite

Quick proof using topological centres (Lau ’86): uniqueinv. meanM

⇒ M ∈Z_`(`∞(G)^∗) =`₁(G)

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Group actions and invariant means

Of course, forGyG: ∃! inv. meanon`∞(G)⇔ G finite Quick proof using topological centres (Lau ’86):

uniqueinv. meanM

⇒ M ∈Z_`(`∞(G)^∗) =`₁(G)