Topological centres for group algebras, actions, and quantum groups

Topological centres for group algebras,actions, and quantum groups

Matthias Neufang

Carleton University (Ottawa)

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

Arens products: Algebraic description

A Banach algebra; as Banach space: A ↪→ A∗∗

∃ 2 canonical extensions of product to A∗∗ (Arens ’51)

X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A

〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉

. . . and the other way around:

〈X3Y , f 〉 = 〈Y , f 3X 〉〈f 3X , a〉 = 〈X , a3f 〉〈a3f , b〉 = 〈f , b · a〉

X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A

〈X2Y , f 〉 = 〈X ,Y 2f 〉

〈Y 2f , a〉 = 〈Y , f 2a〉〈f 2a, b〉 = 〈f , a · b〉

X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A

〈X2Y , f 〉 = 〈X ,Y 2f 〉〈Y 2f , a〉 = 〈Y , f 2a〉

〈f 2a, b〉 = 〈f , a · b〉

X ,Y ∈ A∗∗, f ∈ A∗, a, b ∈ A

Arens products: Topological description

A 3 xi −→ X ∈ A∗∗ (w∗)

A 3 yj −→ Y ∈ A∗∗ (w∗)

X2Y = limi limj xi · yj

X3Y = limj limi xi · yj

2 = 3 ⇔: A Arens 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?

A 3 xi −→ X ∈ A∗∗ (w∗)

A 3 yj −→ Y ∈ A∗∗ (w∗)

X2Y 6= X3Y

A 3 xi −→ X ∈ A∗∗ (w∗)

A 3 yj −→ Y ∈ A∗∗ (w∗)

X2Y 6= X3Y

A 3 xi −→ X ∈ A∗∗ (w∗)

A 3 yj −→ Y ∈ A∗∗ (w∗)

X2Y 6= X3Y

A 3 xi −→ X ∈ A∗∗ (w∗)

A 3 yj −→ Y ∈ A∗∗ (w∗)

X2Y 6= X3Y

A 3 xi −→ X ∈ A∗∗ (w∗)

A 3 yj −→ Y ∈ A∗∗ (w∗)

X2Y 6= X3Y

Topological centres

Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }= { X | Y 7→ X2Y w∗-cont. }

Zr (A∗∗) := { X | Y 2X = Y 3X ∀ Y }= { X | Y 7→ Y 3X w∗-cont. }

A Arens regular :⇔ Z` = Zr = A∗∗

Definition (Dales–Lau ’05)

A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A

Topological centres

Z`(A∗∗) := { X | X2Y = X3Y ∀ Y }

= { X | Y 7→ X2Y w∗-cont. }

Topological centres

A Left Strongly Arens Irregular (LSAI) :⇔ Z` = A

A Right Strongly Arens Irregular (RSAI) :⇔ Zr = AA Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A

Topological centres

A Left Strongly Arens Irregular (LSAI) :⇔ Z` = AA Right Strongly Arens Irregular (RSAI) :⇔ Zr = A

A Strongly Arens Irregular (SAI) :⇔ Z` = Zr = A

Topological centres

Asymmetries

Obviously: Z` = A∗∗ ⇔ Zr = A∗∗However: Z` = A 6⇒ Zr = A

Proposition (Dales–Lau ’05; N. ’05)

LSAI 6⇒ RSAI

Example convolution algebra T (G) = (T (L2(G)), ∗)

ρ ∗ τ :=

LxρLx−1π(τ)(x) dx

G non-compact, second countable ⇒ T (G) LSAI but not RSAI

Asymmetries

Obviously: Z` = A∗∗ ⇔ Zr = A∗∗

However: Z` = A 6⇒ Zr = A

LSAI 6⇒ RSAI

ρ ∗ τ :=

Asymmetries

LSAI 6⇒ RSAI

ρ ∗ τ :=

Asymmetries

LSAI 6⇒ RSAI

ρ ∗ τ :=

Arens products & Kadison–Singer Conjecture

Kadison–Singer Conjecture (’59)

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

Equivalent conjecture about Bessel sequences (Weaver ’04)

∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑

|〈xi , y〉|2 ≤ M ∀ unit vector y ∈ Ck

⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n} with∑i∈Aj

|〈xi , y〉|2 ≤ M − ε ∀ unit vectors y ∈ Ck , ∀ j

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

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

∃ M ≥ 2 and ε > 0 such that:

given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1 and∑i

∃ M ≥ 2 and ε > 0 such that:given x1, . . . , xn ∈ Ck (n ≥ 2) with `2-norm ≤ 1

and∑i

⇒ ∃ partition A1, . . . ,A` (` ≥ 2) of {1, . . . , n}

with∑i∈Aj

Kadison–Singer Conjecture & T (G)

Proposition (N. ’08)

Kadison–Singer Conjecture for discrete G⇒ the map

βG 3 m 7→ m ∈ T (`2(G))∗∗

is multiplicative w.r.t. convolution

This is true for G = Z as well as G = F2 . . .

Transfer of topological dynamics!

Kadison–Singer Conjecture for discrete G

⇒ the mapβG 3 m 7→ m ∈ T (`2(G))∗∗

βG 3 m 7→ m ∈ T (`2(G))∗∗

The Ghahramani–Lau Conjecture

Theorem (Lau–Losert ’88)

L1(G) is SAI for any locally compact group G.

Conjecture (Lau ’94 & Ghahramani–Lau ’95)

M(G) is SAI for any locally compact group G.

Theorem (N. ’05)

The conjecture holds for all non-compact groups G s.t.G has non-measurable cardinality OR k(G) ≥ 2b(G)

One cannot prove in ZFC the existence of measurable cardinals(Ulam ’30)!

Theorem (Losert ’09)

The second condition can be weakened to k(G) ≥ b(G) .

Theorem (N. ’05)

Key technique: Factorization

Definition (N. ’05)

A Banach algebra, κ ≥ ℵ0.

1 A has factorization property of level κ (Fκ) if

∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗

hi = Xi2h (i ∈ I )

2 A has Mazur’s property of level κ (Mκ) ifany X ∈ A∗∗ which is w∗-κ-continuous on A∗, lies in A.

Theorem (N. ’05)

A has Fκ and Mκ for some κ ≥ ℵ0 ⇒ A is SAI

Theorem (N. ’05; Hu–N. ’04)

M(G) has Fk(G) (for non-compact G) and M|G|.

hi = Xi2h (i ∈ I )

Theorem (N. ’05)

∀ (hi )i∈I ⊆ Ball(A∗), |I | ≤ κ

∃ (Xi )i∈I ⊆ Ball(A∗∗) ∃ h ∈ A∗

hi = Xi2h (i ∈ I )

Theorem (N. ’05)

hi = Xi2h (i ∈ I )

Theorem (N. ’05)

hi = Xi2h (i ∈ I )

Theorem (N. ’05)

hi = Xi2h (i ∈ I )

Theorem (N. ’05)

hi = Xi2h (i ∈ I )

Theorem (N. ’05)

Factorization works for any 2nd countable group!

Theorem (N.–Pachl–Steprans ’09)

M(G) is SAI for any locally compact group with |G| ≤ c.In particular, the conjecture holds for all 2nd countable groups.

Idea of proof: Factorization in the dual of singular measures!

Factorization works for any 2nd countable group!

M(G) is SAI for any locally compact group with |G| ≤ c.In particular, the conjecture holds for all 2nd countable groups.

Idea of proof: Factorization in the dual of singular measures!

Separation of singular measures

The following generalizes a result by Prokaj (’03) for G = R.We can assume that G is non-discrete.

For every µ ∈ Ms(G) there exist a Kσ set E ⊆ G and P ⊆ Gs.t. |µ|(G \ E ) = 0, |P| = c, (Ep) ∩ (Ep′) = ∅ if p 6= p′ in P.

Idea Choose compacta Ki with |µ|(G \ Ki ) ≤ 2−i and xi ∈ G“along” a Cantor set C; set E = ∪n ∩∞i=n Ki and construct P from xi and C

Corollary (Separation Lemma)

(Fα)α∈I family of finite subsets of Ms(G) with |I | ≤ c

⇒ ∃ (xα) ⊆ G s.t. (Fα ∗ xα) ⊥ (Fβ ∗ xβ) if α 6= β in I

Factorization in the dual of singular measures

Factorization theorem (N.–Pachl–Steprans ’09)

|G| ≤ c ⇒ ∃ h ∈ BallMs(G)∗ s.t. δGw∗

2h = BallMs(G)∗

Proof.

Sketch

BallMs(G)∗ has w∗-open nhd. basis O with |O| ≤ c (Hu–N. ’06)

O 3 U = { f ∈ BallMs(G)∗ | |〈f , µ〉 − 〈gU , µ〉| < εU ∀ µ ∈ FU }

where FU ⊆ Ms(G) finite, gU ∈ BallMs(G)∗ and εU > 0

Separation Lemma ; xU ∈ G s.t. (FU ∗ xU) ⊥ (FV ∗ xV ) if U 6= V

Define hU ∈ Ms(G)∗ by 〈hU , ν〉 = 〈gU , ν ∗ x−1U 〉

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

2h = BallMs(G)∗

Proof.

Sketch

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

2h = BallMs(G)∗

Proof.

Sketch

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

2h = BallMs(G)∗

Proof.

Sketch

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

2h = BallMs(G)∗

Proof.

Sketch

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

2h = BallMs(G)∗

Proof.

Sketch

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

2h = BallMs(G)∗

Proof.

Sketch

; h ∈ Ms(G)∗ s.t. xU2h ∈ U

Solution of the conjecture

Let G be a locally compact group with |G| ≤ c.Then M(G) is SAI.

Proof.

Let m ∈ Z`(M(G)∗∗)

⇒ ms := m |Ms(G)∗ is w∗-cont. on any set of the form

δGw∗

2h ⊆ Ms(G)∗ where h ∈ Ms(G)∗

Factorization theorem ⇒ ms is w∗-cont. on BallMs(G)∗

⇒ ms ∈ M(G)

ma := m |L1(G)∗∈ L1(G)∗∗ satisfies ma = m −ms ∈ Z`(M(G)∗∗)

⇒ ma ∈ Z`(L1(G)∗∗) = L1(G), and m = ma + ms ∈ M(G) .

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Proof.

Let m ∈ Z`(M(G)∗∗)

δGw∗

⇒ ms ∈ M(G)

Beyond second countability

Let G be a product of at most c many compact, metrizable groups.Then M(G) is SAI.

G. Dales, T. Lau & D. Strauss (’09)Second duals of measure algebras

The dual setting

Problem (Cechini–Zappa ’81)

Is A(G) SAI?

Yes for large classes of amenable groups.

Theorem (Losert ’02 & ’04)

No for G = F2 and also for G = SU(3) !

Theorem (Filali–Monfared–N. ’08)

Yes for any compact group that is sufficiently non-metrizable(b(G) has uncountable cofinality); e.g., SU(3)ℵ1 and SU(3)c

Yes for SU(3)ℵ0

The dual setting

Is A(G) SAI?

Yes for SU(3)ℵ0

The dual setting

Is A(G) SAI?

Yes for SU(3)ℵ0

The dual setting

Is A(G) SAI?

Yes for SU(3)ℵ0

The dual setting

Is A(G) SAI?

Yes for SU(3)ℵ0

The dual setting

Is A(G) SAI?

Yes for SU(3)ℵ0

Method of proof:Factorization & Mazur’s property for A(G)

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

∀ (Tα)α∈I ⊆ BallL(G) with |I | ≤ b(G)

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

So A(G) has (a slightly weakened form of) Fb(G).

Theorem (Hu–N. ’04)

A(G) has Mb(G)·ℵ0.

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

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

∃ (X kα )α∈I ⊆ BallL(G)∗ (k = 1, . . . , n)

∃ T k ∈ L(G) (k = 1, . . . , n) s.t.

Tα =n∑

X kα 2 T k

Strategy of (technical) proof

Tα =n∑

X kα 2 T k

use special central projections in L(G) arising from family ofcompact normal subgroups of G decreasing to e, of sizeI = b(G)

double the index set I (“family breeding”): α ; (α, β)

“separate” projections by coordinate functions of ab(G)-family of irred. unitary rep.s of Gthis gives a family of pairwise orthogonal projections in L(G)

truncate Tα with these and sum up (w∗) ; operators T k

X kα are w∗-cluster points in A(G)∗∗ of rep.s along β

Tα =n∑

X kα 2 T k

Tα =n∑

X kα 2 T k

Tα =n∑

X kα 2 T k

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

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

Tα =n∑

X kα 2 T k

Tα =n∑

X kα 2 T k

Tα =n∑

X kα 2 T k

Topological centres and multipliers

Problem (Lau–Ulger ’96)

A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .Are Ker(X2) and X2(BallA∗) w∗-closed?

Theorem (Hu–N.–Ruan ’09)

No for A = A(SU(3))

Proof: Combine Losert’s result Z (A∗∗) 6= A with the following

Assume A separable. Then, for X ∈ Zr (A∗∗):

X ∈ A ⇔ Ker(X2) and X2(BallA∗) are w∗-closed

This relies on work by Godefroy–Talagrand ’89 & N. ’01

A Banach algebra with BAI s.t. A∗ vN algebra. Let X ∈ Zr (A∗∗) .Consider X2 : A∗ 3 h 7→ X2h ∈ A∗ .

Are Ker(X2) and X2(BallA∗) w∗-closed?

No for A = A(SU(3))

Ghahramani–Farhadi’s multiplier problem

Problem (Duncan–Hosseiniun ’79)

G locally compact group. Does the involution on L1(G) extend toan involution on its bidual?

Proposition (Farhadi–Ghahramani ’07)

1 This fails for non-discrete groups.

2 It also fails for all groups with the following property (∗):Consider any Φ : L∞(G)∗∗ → L∞(G)∗∗ normal & surjective;if Φ commutes with L1(G), then also with L1(G)∗∗.

Problem (Farhadi–Ghahramani ’07)

Does every group G satisfy (∗) ?

Solution to the multiplier problem

Theorem (N. ’08)

The problem has a negative answer for all infinite countablediscrete abelian groups.

For the proof, consider βG ⊆ `1(G)∗∗ :

compact right topological semigroup with first Arens product

Growth/Remainder/Corona:

G∗ := βG \ G

⇒ G∗ compact right topological semigroup

Theorem (N. ’08)

G∗ := βG \ G

Theorem (N. ’08)

G∗ := βG \ G

Theorem (N. ’08)

G∗ := βG \ G

Module maps

m ∈ βG is called left cancellable if λm is injective on βG

Proposition (Dales–Lau–Strauss ’08)

m ∈ βG left cancellable ⇒ λm : `1(G)∗∗ → `1(G)∗∗ isometry

Write A := `1(G)∗∗.

∃ m ∈ G∗ ⊆ A such that m is left cancellable in βGProposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective

Need to show:

1 Φ is a right `1(G)-module map

2 Φ is not a right `1(G)∗∗-module map

Module maps

Write A := `1(G)∗∗.

Need to show:

Module maps

Write A := `1(G)∗∗.

∃ m ∈ G∗ ⊆ A such that m is left cancellable in βG

Proposition ⇒ Φ := λ∗m : A∗ → A∗ (normal &) surjective

Need to show:

Module maps

Write A := `1(G)∗∗.

Need to show:

Module maps

Write A := `1(G)∗∗.

Need to show:

Φ = λ∗m is a right `1(G)-module map

Recall: A = `1(G)∗∗

∀ H ∈ A∗, a ∈ `1(G) ⊆ A, b ∈ A

〈Φ(H2a), b〉 = 〈H, a ∗m ∗ b〉

But a ∈ `1(G) = Z (A), so a commutes with m ∈ G∗ ⊆ A :

〈Φ(H2a), b〉 = 〈H,m ∗ a ∗ b〉 = 〈Φ(H)2a, b〉

as desired.

∀ H ∈ A∗, a ∈ `1(G) ⊆ A, b ∈ A

〈Φ(H2a), b〉 = 〈H, a ∗m ∗ b〉

〈Φ(H2a), b〉 = 〈H,m ∗ a ∗ b〉 = 〈Φ(H)2a, b〉

as desired.

∀ H ∈ A∗, a ∈ `1(G) ⊆ A, b ∈ A

〈Φ(H2a), b〉 = 〈H, a ∗m ∗ b〉

〈Φ(H2a), b〉 = 〈H,m ∗ a ∗ b〉 = 〈Φ(H)2a, b〉

as desired.

Φ = λ∗m is not a right `1(G)∗∗-module map

Suppose Φ is a right A-module map

⇒ ∀ H ∈ A∗, a, b ∈ A

〈H, a ∗m ∗ b〉 = 〈Φ(H2a), b〉 = 〈Φ(H)2a, b〉 = 〈H,m ∗ a ∗ b〉

⇒ a ∗m ∗ b = m ∗ a ∗ b ∀ a, b ∈ A⇒ (with b = δe) m ∈ Z (A) = `1(G)

This contradicts m ∈ G∗.

⇒ ∀ H ∈ A∗, a, b ∈ A

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 only invariant mean onL∞(Sn) for the O(n + 1)-action.

What about the discrete situation?

Of course, for G y G: ∃! inv. mean on `∞(G) ⇔ G finite

Quick proof using topological centres (Lau ’86):

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

⇒ M finite Haar measure, so G finite!

What about general actions G y X ?

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

unique inv. mean M

⇒ M ∈ Z`(`∞(G)∗) = `1(G)

An independence result for general actions

Theorem (Foreman ’94)

The statement “∃ locally finite group G of permutations of N witha unique invariant mean on `∞(N)” is independent of ZFC!

CH ⇒ ∃ locally finite group of permutations of N, of size c,with a unique invariant mean on `∞(N)

Theorem (Rosenblatt–Talagrand ’81)

Infinite countable groups never admit a unique invariant mean.

How many?

G y X with G,X infinite countable.G amenable ⇒ ∃ 2c many invariant means on `∞(X )

How many?

Arens type product and topological centrefor group actions

Definition

G y X .

1 For x ∈ βX and h ∈ `∞(X ) = C (βX ) define x2h ∈ `∞(G) by

(x2h)(g) := h(gx)

2 Define a “convolution” `∞(G)∗ × βX → `∞(X )∗ by

〈m2x , h〉 := 〈m, x2h〉

Definition

Zt(G,X ) := { m ∈ `∞(G)∗ | βX 3 x 7→ m2x w∗-cont. }

Definition

G y X .

(x2h)(g) := h(gx)

〈m2x , h〉 := 〈m, x2h〉

Definition

G y X .

1 For x ∈ βX and h ∈ `∞(X ) = C (βX )

define x2h ∈ `∞(G) by

(x2h)(g) := h(gx)

〈m2x , h〉 := 〈m, x2h〉

Definition

G y X .

(x2h)(g) := h(gx)

〈m2x , h〉 := 〈m, x2h〉

Definition

G y X .

(x2h)(g) := h(gx)

〈m2x , h〉 := 〈m, x2h〉

Definition

G y X .

(x2h)(g) := h(gx)

〈m2x , h〉 := 〈m, x2h〉

Definition

Foreman’s group has non-trivial topological centre!

G y X with G amenable and Zt(G,X ) = `1(G).If the number of inv. means on `∞(X ) is finite, then G is finite.

Corollary (N.–Pachl–Steprans ’09)

CH ⇒ Zt(G,X ) 6= `1(G) for Foreman’s group!

But we (N.–P.–S.) have a class of infinite groups of size < c withZt(G,N) = `1(G)

G y X with G amenable and Zt(G,X ) = `1(G).

If the number of inv. means on `∞(X ) is finite, then G is finite.

Topological centres for quantum group algebras

Definition

Hopf–von Neumann algebra (M, Γ)

M von Neumann algebra

Γ : M → M⊗M co-multiplication

Examples

M = L∞(G) = L1(G)∗

Γ = adjoint of convolution product ∗

M = L(G) = A(G)∗

Γ = adjoint of pointwise product •

Definition

Examples

M = L∞(G) = L1(G)∗

M = L(G) = A(G)∗

Definition

Examples

M = L∞(G) = L1(G)∗

M = L(G) = A(G)∗

Definition

Examples

M = L∞(G) = L1(G)∗

M = L(G) = A(G)∗

Locally compact quantum groups

Non-commutative integration

N.s.f. weight λ : M+ → [0,∞]

Mλ := lin { x ∈ M+ | λ(x) <∞ }

Definition (Kustermans–Vaes ’00)

LC Quantum Group G = (M, Γ, λ, ρ)

λ left Haar weight on M:

λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ

ρ right Haar weight on M:

ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ

Theorem (Kustermans–Vaes ’00)

“Pontryagin duality”G ∼= G

N.s.f. weight λ : M+ → [0,∞]

Mλ := lin { x ∈ M+ | λ(x) <∞ }

λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ

ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ

N.s.f. weight λ : M+ → [0,∞]

Mλ := lin { x ∈ M+ | λ(x) <∞ }

λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ

ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ

N.s.f. weight λ : M+ → [0,∞]

Mλ := lin { x ∈ M+ | λ(x) <∞ }

λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ

ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ

N.s.f. weight λ : M+ → [0,∞]

Mλ := lin { x ∈ M+ | λ(x) <∞ }

λ((f ⊗ id)Γx) = 〈f , 1〉 λ(x) ∀ f ∈ M∗, x ∈ Mλ

ρ((id⊗ f )Γx) = 〈f , 1〉 ρ(x) ∀ f ∈ M∗, x ∈ Mρ

Algebras over quantum groups

L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)

L1(G) Banach algebra via f ∗ g = Γ∗(f ⊗ g)

LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)

WAP(G) := { T ∈ L∞(G) | L1(G) 3 f 7→ T2f weakly compact }

C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖

where W left fundamental unitary: Γ(x) = W ∗(1⊗ x)W

L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)

LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)

C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖

L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)

LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)

C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖

L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)

LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)

C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖

L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)

LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)

C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖

L∞(G) := M L1(G) := M∗ L2(G) := L2(M, λ)

LUC(G) := lin L∞(G)2L1(G) ⊆ L∞(G)

C0(G) := { (id⊗ τ)(W ) | τ ∈ T (L2(G)) }‖·‖

Characterization of discrete quantum groups

Assume G co-amenable (i.e., L1(G) has BAI) s.t. L1(G) separable.TFAE:

G discrete (i.e., L1(G) unital)

LUC(G) = L∞(G)

Assume G co-amenable (i.e., L1(G) has BAI) s.t. L1(G) separable.

LUC(G) = L∞(G)

Assume G co-amenable (i.e., L1(G) has BAI) s.t. L1(G) separable.TFAE:

LUC(G) = L∞(G)

Characterization of compact quantum groups

Since LUC(G) ⊆ L∞(G) we have

LUC(G)∗ ←− L1(G)∗∗

; Transport of left Arens product ; LUC(G)∗ Banach algebra

Zt(LUC(G)∗) := { X ∈ LUC(G)∗ | Y 7→ X2Y w∗-cont. }

G compact (i.e., C0(G) unital)

LUC(G) ⊆WAP(G) and Zt(LUC(G)∗) = M(G)

Question G = L(G) with G discrete?⇒ WAP(G) ⊆ LUC(G)

If yes, then there is NO infinite G with A(G) Arens regular!Open for Olshanskii group . . .

LUC(G)∗ ←− L1(G)∗∗

Characterizations using invariant meanson quantum groups

Definition

G amenable :⇔ ∃ mean on L∞(G) s.t.

f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)

Let G be amenable with L1(G) separable or SAI.Then: G uniquely amenable ⇔ G compact

Let G be amenable & co-amenable, with L1(G) separable.Then: L1(G) Arens regular ⇔ G finite

Definition

f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)

Definition

f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)

Let G be amenable with L1(G) separable or SAI.

Then: G uniquely amenable ⇔ G compact

Definition

f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)

Definition

f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)

Let G be amenable & co-amenable, with L1(G) separable.

Then: L1(G) Arens regular ⇔ G finite

Definition

f 2M = 〈f , 1〉 M ∀ f ∈ L1(G)

From quantum groups back to groups

McbL1(G) := { Φ : L1(G)→ L1(G) | Φ CB, Φ(a∗b) = a∗Φ(b) }

Consider m ∈McbL1(G) s.t.

m∗ UCP (= Markov operator)

m invertible and complete isometry on L1(G)

Denote by G the set of those multipliers.

Examples

L∞(G) ∼= G

L(G) ∼= G

Examples

L∞(G) ∼= G

L(G) ∼= G

Examples

L∞(G) ∼= G

L(G) ∼= G

Examples

L∞(G) ∼= G

L(G) ∼= G

Examples

L∞(G) ∼= G

L(G) ∼= G

Examples

L∞(G) ∼= G

L(G) ∼= G

Examples

L∞(G) ∼= G

L(G) ∼= G

A functor LC quantum groups → LC groups

Theorem (Kalantar–N. ’09)

G is a LC group w.r.t. point weak topology on L1(G)

G ∼= Sp(L1(G))

The functor G→ G preserves

local compactness

compactness

discreteness

(hence) finiteness

The functor also “almost commutes” with quantum group duality.

G ∼= Sp(L1(G))

local compactness

compactness

discreteness

(hence) finiteness

G ∼= Sp(L1(G))

local compactness

compactness

discreteness

(hence) finiteness

G ∼= Sp(L1(G))

local compactness

compactness

discreteness

(hence) finiteness

G ∼= Sp(L1(G))

local compactness

compactness

discreteness

(hence) finiteness

Structure of G

G compact matrix pseudogroup (Woronowicz ’87)⇒ G is a compact Lie group

Example Woronowicz’s SUq(2) with deformation parameterq ∈ (0, 1]

SUq(2) = C (SU(2)) for q = 1

Non-commutative C ∗-algebra for q ∈ (0, 1)

SUq(2) ∼= SU(2) for q = 1

SUq(2) ∼= T for q 6= 1

G compact, non-Kac with L1(G) separable ⇒ G uncountable

Structure of G

SUq(2) = C (SU(2)) for q = 1

SUq(2) ∼= SU(2) for q = 1

SUq(2) ∼= T for q 6= 1

Structure of G

SUq(2) = C (SU(2)) for q = 1

SUq(2) ∼= SU(2) for q = 1

SUq(2) ∼= T for q 6= 1

Structure of G

SUq(2) = C (SU(2)) for q = 1

SUq(2) ∼= SU(2) for q = 1

SUq(2) ∼= T for q 6= 1

Structure of G

SUq(2) = C (SU(2)) for q = 1

SUq(2) ∼= SU(2) for q = 1

SUq(2) ∼= T for q 6= 1

Structure of G

SUq(2) = C (SU(2)) for q = 1

SUq(2) ∼= SU(2) for q = 1

SUq(2) ∼= T for q 6= 1

Heisenberg relation for quantum groups

G abelian. For s ∈ G and γ ∈ G

Ls Mγ = 〈γ, s〉︸︷︷︸∈T

Mγ Ls

We obtain a generalization to quantum groups, using acommutation result by Junge–N.–Ruan (’09):

Theorem: Non-commutative Torus (Kalantar–N. ’09)

g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.

g g = 〈g , g〉 g g⟨˜G, G

⟩=: G0 is a subgroup of T .

Example SUq(2)0 = T (q 6= 1)

Ls Mγ = 〈γ, s〉︸︷︷︸∈T

Mγ Ls

g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.

g g = 〈g , g〉 g g⟨˜G, G

Ls Mγ = 〈γ, s〉︸︷︷︸∈T

Mγ Ls

g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.

g g = 〈g , g〉 g g⟨˜G, G

Ls Mγ = 〈γ, s〉︸︷︷︸∈T

Mγ Ls

g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.

g g = 〈g , g〉 g g

⟨˜G, G

Ls Mγ = 〈γ, s〉︸︷︷︸∈T

Mγ Ls

g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.

g g = 〈g , g〉 g g⟨˜G, G

Ls Mγ = 〈γ, s〉︸︷︷︸∈T

Mγ Ls

g ∈ G, g ∈ ˜G ⇒ ∃ 〈g , g〉 ∈ T s.t.

g g = 〈g , g〉 g g⟨˜G, G

Semigroup compactifications from quantum groups

Assume G co-amenable.

McbL1(G) ∼= M(G) ↪→ Zt(LUC(G)∗)

The embedding G ⊆McbL1(G) in LUC(G)∗ gives rise to

GLUC := Gw∗

Then GLUC is a compact right topological semigroup.

Note: G = L∞(G) for a LC group G ⇒ GLUC = GLUC

GLUC := Gw∗

Unification via T (L2(G))

Co-multiplications Γ and Γ extend to

B(L2(G)) → B(L2(G)) ⊗ B(L2(G))

⇒ Γ∗ = m and Γ∗ = m yield 2 dual products

T (L2(G)) ⊗ T (L2(G)) → T (L2(G))

m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)

Here, σ(ϕ⊗ τ) = τ ⊗ ϕ is the flip.

Duality = Anti-Commutation Relation on tensor level

B(L2(G)) → B(L2(G)) ⊗ B(L2(G))

T (L2(G)) ⊗ T (L2(G)) → T (L2(G))

m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)

B(L2(G)) → B(L2(G)) ⊗ B(L2(G))

T (L2(G)) ⊗ T (L2(G)) → T (L2(G))

m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)

B(L2(G)) → B(L2(G)) ⊗ B(L2(G))

T (L2(G)) ⊗ T (L2(G)) → T (L2(G))

m ◦ (m⊗ id) = m ◦ (m⊗ id) ◦ (id⊗ σ)

T (L2(G))as a home for convolution and pointwise product

T (L2(G)) ∼= T (L2(G))

L1(G) L1(G)

∗ •

On T (L2(G)) we can compare“convolution” and “pointwise product”!

(ϕ ∗ τ) • ψ = (ϕ • ψ) ∗ τ(Kalantar–N. ’08)

T (L2(G)) ∼= T (L2(G))

L1(G) L1(G)

∗ •

T (L2(G)) ∼= T (L2(G))

L1(G) L1(G)

∗ •

Some cohomology for LC quantum groups

The following generalizes results by Pirkovskii on (T (L2(G)), ∗) toquantum groups.

L1(G) is projective in mod–T (L2(G))

⇔ L1(G) has the Radon–Nikodym Property

C is projective in mod–T (L2(G))

⇔ G is compact

Topological centres for group algebras, actions, and quantum groups

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