Abstract Harmonic Analysis and Related Operator Algebras...

Abstract Harmonic Analysis and Related

Operator Algebras on Locally Compact

Quantum Groups

Zhong-Jin Ruan

University of Illinois at Urbana-Champaign

ICM Satellite Conference on Operator Algebras and Applications

August 8-12, 2014

Cheongpung, Korea

1

In this talk, we plan to discuss

• Historical Backgroup of Locally Compact Quantum Groups

• Abstract Quantum Harmonic Analysis

• Amenability and Approximation Properties of Quantum Groups

• Exotic Quantum Group C*-algebras

2

Historical Background

3

Pontryagin Duality (1934)

Let G be a locally compact abelian group. There exists a dual group

G = {γ : G→ T : continuous homomorphisms}.

G has a canonical abelian group structure and we have the homeomor-

phism

ˆG ∼= G.

In particular, we have the Plancherel theorem

L2(G) = L2(G).

How to generalize such duality results to non-abelian groups ?

4

Fourier transformation gives injective *-homomorphism

λ : L1(G)→ C0(G) ⊆ L∞(G)

This suggests that we can consider the corresponding group C*-algebras

and group von Neumann algebras as the duality !

5

Group C*-algebras and Group von Neumann Algebras

Let G be a locally compact group and µ be the left Haar measure of G.

The left regular representation of G

λ : s ∈ G→ λs ∈ B(L2(G))

is given by

λs(ξ)(t) = ξ(s−1t).

We get

• reduced group C*-algebra C∗λ(G) = {λ(f) : f ∈ L1(G)}−‖·‖ ⊆ B(L2(G))

• the left group von Neumann algebra L(G) = {λs}′′ ⊆ B(L2(G)).

6

We get a natural “duality” between

L∞(G) and L(G)

or

C0(G) and C∗λ(G)

since if G is an abelian group with dual group G, we have

L(G) = L∞(G) and C∗λ(G) = C0(G)

7

How do we recover the group structure from L∞(G) or L(G) ?

8

Examples

For the Dihedral group D4 or the quaternion group Q (of order 8), we

have

`∞(D4) = `∞(8) = `∞(Q)

and

L(D4) = C⊕ C⊕ C⊕ C⊕M2(C) = L(Q).

We can not recover the group structure from their algebraic structure.

9

Co-multiplication on L∞(G)

To recover the group structure from L∞(G), we need to consider a

co-multiplication

Γa : f ∈ L∞(G)→ Γa(f) ∈ L∞(G)⊗L∞(G)

given by

Γa(f)(s, t) = f(st).

It is easy to see the Γa is a normal injective *-homomorphism such that

it satisfies the co-associativity condition

(Γa ⊗ ι)Γa = (ι⊗ Γa)Γa

i.e. we have

(Γa ⊗ ι)Γa(f)(s, t, u) = f((st)u) = f(s(tu)) = (ι⊗ Γa)Γa(f)(s, t, u).

We call (L∞(G),Γa) a commutative Hopf von Neumann algebra.

10

Co-involution on L∞(G)

To recover the inverse of G, we can consider a co-involution

κa(h) = h

which is a weak* continuous *-anti-automorphsim on L∞(G) such that

Γa ◦ κa = Σ(κa ⊗ κa) ◦ Γa,

where Σ(a⊗ b) = b⊗ a is the flip map.

Moreover, we can use the left Haar measure to obtain a normal faithful

weight

ϕa : h ∈ L∞(G)+ → ϕa(h) =∫Gh(s)ds ∈ [0,+∞]

on L∞(G) which satisfies (ι⊗ ϕa)Γ(h) = ϕa(h)1 since

(ι⊗ ϕa)Γ(h)(s) =∫Gh(st)dt = (

∫Gh(t)dt) = ϕa(h).

So we call ϕa a left Haar weight on L∞(G).

This gives a Kac algebra structure on L∞(G) !

11

Kac Algebras

G. Kac introduced Kac algebras K = (M,Γ, κ, ϕ), for unimodular case,in the 60’s.

The theory was completed for general (non-unimodular) case in the70’s by two groups: Kac-Vainerman in Ukraine and Enock-Schwartz inFrance (see Enock-Schwartz’s book 1992).

There exists a perfect Pontryagin duality

ˆK = K.

Examples:

• Commutative Kac algebras: Ka(G) = (L∞(G),Γa, κa, ϕa)

• Co-commutative Kac algebras: Ks(G) = (L(G),ΓG, κG, ϕG)

We have the natural duality

Ka(G) = Ks(G) and Ks(G) = Ka(G).

12

Kac Algebra Structure on L(G)

There is a natural co-associative co-multiplication

ΓG : L(G)→ L(G)⊗L(G)

on L(G) given by

ΓG(λ(s)) = λ(s)⊗ λ(s).

This is co-commutative in the sense that

Σ ◦ ΓG = ΓG.

To recover the inverse, we consider a co-involution κG on L(G) givenby

κG(λ(s)) = λ(s−1).

Moreover, we can obtain a normal faithful (Plancherel) weight ϕG onL(G). So (L(G),ΓG, κG, ϕG) is a co-commutative Kac algebra.

Remark If G is a discrete group, then

ϕG(x) = ψG(x) = 〈xδe|δe〉.

13

Locally Compact Quantum Groups (LCQG)

In 1987, Woronowicz introduced SUq(2,C), a natural quantum defor-

mation of SU(2,C). He showed that SUq(2,C) does not correspond to

any Kac algebra since the “co-involution” is unbounded on SUq(2,C).

Since then, several different definitions of LCQG have been investigated:

• Baaj and Skandalis 1993: Regular Multiplicative Unitaries

• Woronowicz 1996: Manageable Multiplicative Unitaries

• Kustermans and Vaes 2000: Quantum Groups, C∗-algebra setting

• Kustermans and Vaes: 2003: Quantum Groups, von Neumann algebra

setting.

15

Kustermans and Vaes’ Definition

A LCQG is G = (M,Γ, ϕ, ψ) consisting of

(1) a von Neumann algebra M

(2) a co-multiplication Γ : M → M⊗M , i.e. a unital normal *-homomorphism satisfying the co-associativity condition

(id⊗ Γ) ◦ Γ = (Γ⊗ id) ◦ Γ.

(3) a left Haar wight ϕ, i.e. a n.f.s weight ϕ on M satisfying

(ι⊗ ϕ)Γ(x) = ϕ(x)1

(4) a right Haar weight ψ, i.e. n.f.s weight ψ on M satisfying

(ψ ⊗ ι)Γ(x) = ψ(x)1.

It is known that for every locally compact quantum group G = (M,Γ, ϕ, ψ),there exists a dual quantum group G = (M, Γ, ϕ, ψ) such that we mayobtain the perfect Pontryagin duality

ˆG = G.16

Every Kac algebra K = (M,Γ, κ, ϕ) is a LCQG since we can obtain the

right Haar weight

ψ = ϕ ◦ κ

via ϕ and κ !

17

Abstract Quantum Harmonic Analysis

18

Quantum Convolution Algebras (L1(G), ?)

Let G = (L∞(G),Γ, ϕ, ψ) be a locally compact quantum group. Here we

use the notion L∞(G) for the quantum group von Neumann algebra M .

The co-multiplication

Γ : L∞(G)→ L∞(G)⊗L∞(G)

induces an associative completely contractive multiplication

? = Γ∗ : f1 ⊗ f2 ∈ L1(G)⊗L1(G)→ f1 ? f2 = (f1 ⊗ f2) ◦ Γ ∈ L1(G)

on L1(G) = L∞(G)∗.

Remark: In general, we do not have an involution on L1(G) since the

antipode S is unbounded on L∞(G). But we have a dense *-subalgebra

L]1(G) = {ω ∈ L1(G) : ∃ ω] ∈ L1(G) such that ω] = ω∗ ◦ S}.

For Kac algebras, we have L1(G) = L]1(G) and the involution is com-

pletely isometric.

19

If G = (L∞(G),Γa, ϕa, ψa) is a commutative LCQG, then ? = Γ∗ is just

the multiplication on the convolution algebra L1(G) = L1(G).

If G = (L(G),ΓG, ϕG, ψG) is a co-commutative LCQG, then ? = ΓG∗ is

just the pointwise multiplication on the Fourier algebra

L1(G) = L(G)∗ = A(G).

We note that Fourier algebra was first studied by Eymard in 1964. It

can be defined to be the space

A(G) = {f : G→ C : continuous such that f(s) = 〈λ(s)ξ|η〉}

It was shown by Eymard in 1964 that A(G) is a commutative Banach

algebra with the pointwise multiplication and norm

‖f‖A(G) = inf{‖ξ‖‖η‖}.

We have a natural operator space structure on A(G), regarded as the

operator predual of L(G).

20

Representations

A representation of G is a non-degenerate c.c. homomorphism π :L1(G)→ B(H) such that π restricted to L

]1(G) is a *-homomorphism.

Each representation (π,H) is uniquely associated with a unitary operatorUπ ∈M(C0(G)⊗K(H)) ⊆ L∞(G)⊗B(H) such that

π(ω) = (ω ⊗ ι)(Uπ) (ω ∈ L1(G))

In particular, we have the left regular representation

λ : ω ∈ L1(G)→ λ(ω) = (ω ⊗ ι)(W ) ∈ Cλ(G) = C0(G) ⊆ L∞(G) ⊆ B(L2(G))

and the universal representation

πu : ω ∈ L1(G)→ πu(ω) = (ω ⊗ ι)(W) ∈ C∗(G) = Cu(G) ⊆ B(Hu)

Here W is the left fundamental unitary of G and W is Kustermans semi-universal version of W . If G is a finite group, we can explicitly write

W =∑s∈G

δs ⊗ λ(s) and W =∑s∈G

δs ⊗ πu(s).

21

Summary

We have the following quantum group algebras:

L∞(G) and L∞(G) = L(G)

C0(G) and C0(G) = C∗λ(G)

Cu(G) and Cu(G) = C∗(G)

L1(G) ⊇ L]1(G) and L]1(G) ⊆ L1(G) = A(G)

Here C0(G) = C∗λ(G) is the reduced quantum group C*-algebra and

C∗u(G) = C∗(G) is the universal/full quantum group C*-algebra.

22

Quantum Fourier-Stieltjes Algebra B(G) = Cu(G)∗

Let G be a locally compact quantum group and let G be its quantum

dual. There is a natural co-multiplication

Γu : Cu(G)→M(Cu(G)⊗ Cu(G))

on the universal (or full) quantum group C*-algebra Cu(G) = C∗(G).

Taking the adjoint, we obtain a completely contractive multiplication

on Cu(G)∗ given by

µ ? ν = (µ⊗ ν) ◦ Γu (µ, ν ∈ Cu(G)∗).

In this case, we call B(G) = Cu(G)∗ to be the quantum Fourier-Stieltjes

algebra of G. It contains A(G) = L1(G) as a norm closed two-sided

ideal.

Given µ in B(G), we can define two completely bounded maps

mlµ(f) = µ ? f and mr

µ(f) = f ? µ

on L1(G). In fact these maps are completely bounded left (resp. right)

centralizers of L1(G).

23

Completely Bounded/Herz-Schur Multipliers of Groups

A function ϕ : G→ C is a multiplier of G if the multiplication map

mϕ : f ∈ A(G)→ ϕf ∈ A(G)

is well defined on A(G). Such a map is automatically bounded.

A function ϕ : G → C is a completely bounded multiplier of G if mϕ iscompletely bounded on A(G).

Remarks: Completely bounded multipliers ϕ are contained in Cb(G) =M(C0(G)) and can be characterized by the following Gilbert theorem.

Theorem: A continuous function ϕ : G → C is a cb-multiplier with‖mϕ‖cb ≤ C if and only if there exist bounded continuous maps α =[αi], β = [βi] : G→ `2(I) for some Hilbert space `2(I) such that sup{‖α(t)‖} sup{‖β(s)‖} ≤C and

ϕ(st−1) = β∗(s)α(t) =∑i

βi(s)αi(t).

In this case, we also call ϕ a Herz-Schur multiplier.

24

Completely Bounded/Herz-Schur Multipliers of Quantum Groups

Let G be a locally compact quantum group.

•We first expect that a Herz-Schur multiplier of G should be an element

in Cb(G) = M(C0(G)) ⊆ L∞(G) .

• Then we expect that it is associated with a completely bounded left

(right, or double) centralizer on A(G) = L1(G).

• Finally, we expect that it could have the corresponding Gilbert theo-

rem.

This was first studied by Kraus and R around 1996, for completely

bounded double multipliers of Kac algebras. Later on, these results

have generalized to all cases for locally compact quantum groups by

Junge-Neufang-R, Hu-Neufang-R (around 2009-11) and Daws (2011-

12).

25

Let us consider the left multiplier case and we recall W ∈ L∞(G)⊗L∞(G).

Theorem : [Kraus-R 1996, Junge-Neufang-R 2009] Let S be acompletely bounded map on L1(G) = A(G). TFAE:

1. S is a completely bounded left centralizer , i.e. S(f ? g) = S(f) ? g,on L1(G). Then taking the adjoint, we get a normal cb-map S∗ onL∞(G) such that

Γ(S∗(x)) = (S∗ ⊗ ι)Γ(x) for x ∈ L∞(G).

2. There exists a unique normal completely bounded L∞(G)′-bimodulemap Φ on B(L2(G)) such that

W ∗(1⊗ Φ(x))W = (S∗ ⊗ ι)(W ∗(1⊗ x)W ) for x ∈ B(L2(G)).

In this case, we have Φ|L∞(G) = S∗.

3. There exists a unique operator a ∈ L∞(G) such that

(S∗ ⊗ ι)(W ) = (1⊗ a)W or (Φ⊗ ι)(W ) = (1⊗ a)W

It was shown by Daws that the operator a ∈ L∞(G) is actually containedin M(C0(G)). We call a the Herz-Schur multiplier of G. In this case,we say a ∈M l

cb(L1(G)).

26

Remark 1: Since the left regular representation of G is given by

λ(f) = (f ⊗ ι)(W ),

condition 3

(S∗ ⊗ ι)(W ) = (1⊗ a)W

implies that for any f ∈ L1(G),

aλ(f) = (f ⊗ ι)((1⊗ a)W ) = (f ⊗ ι)((S∗ ⊗ ι)(W )) = λ(S(f)).

In this case, we get

λ(f)→ aλ(f) = λ(S(f))

and thus

S(f) = λ−1(aλ(f))

is a completely bounded left multiplier map (or a left centralizer) of

L1(G).

27

A Gilbert Characterization for Herz-Schur Multipliers

Theorem : [Daws 2012] An element a ∈ Cb(G) is a Herz-Schur multiplier

if and only if there exist bounded elements [αi], [βi] ∈MI,1(L∞(G)) such

that ∑i

(1⊗ β∗i )Γ(αi) = a⊗ 1.

Daws used a Hilbert module approach. Actually, this result is an imme-

diate consequence of Condition 3 in above theorem.

28

Theorem : [Daws 2012] An element a ∈ Cb(G) is a Herz-Schur multiplier

if and only if there exist bounded elements [αi], [βi] ∈MI,1(L∞(G)) such

that ∑i

(1⊗ β∗i )Γ(αi) = a⊗ 1.

Proof: Let a be a completely bounded left multiplier. By condition 3,

there exists a normal completely bounded L∞(G)′-bimodule map Φ on

B(L2(G)) such that (Φ⊗ ι)(W ) = (1⊗ a)W . Since W = ΣW ∗Σ, we get

(ι⊗ Φ)(W ∗) = (a⊗ 1)W ∗ or (ι⊗ Φ)(W ∗)W = a⊗ 1

Now by Haagerup’s result, the normal L∞(G)’-bimodule map Φ has the

expression Φ(x) =∑i β∗xαi for some [αi], [βi] ∈ MI,1(L∞(G)). There-

fore, ∑i

(1⊗ β∗i )Γ(αi) = (ι⊗ Φ)(W ∗)W = a⊗ 1.

29

Positive Definite Herz-Schur Multipliers

We note that L1(G) = A(G) is an ideal in B(G) = Cu(G)∗ and the left

regular representation

λ : f ∈ L1(G)→ (f ⊗ ι)(W ) = (ι⊗ f)(W ∗) ∈ C0(G) ⊆ L∞(G)

extends naturally to a representation

λ : µ ∈ B(G)→ (ι⊗ µ)(W∗) ∈ Cb(G) = M(C0(G)).

It is easy to see that for every µ ∈ B(G), a = λ(µ) is a Herz-Schur

multiplier. The converse is not true in general (even for groups).

Theorem: [Daws 2012] A Herz-Schur multiplier a is positive definite

if and only if a = λ(µ) for some positive µ ∈ B(G)+.

A completely bounded/Herz-Schur multiplier a is positive definite if the

corresponding map Φ is completely positive.

30

Amenability and Approximation Properties of Quantum Groups

31

Amenability

Let G be a locally compact quantum group.

• G is amenable if there exists a state m : L∞(G)→ C such that

(ι⊗m)Γ(x) = m(x)1 (x ∈ L∞(G)).

• G is co-amenable if L1(G) has a bounded approximate identity. Thisis equivalent to Cu(G) = C0(G).

Remark: For general locally compact quantum groups, we have

G is co-amenable ⇒ G is amenable ⇒ L∞(G) is injective.

Theorem: [R 1996] All three statements are equivalent if G is adiscrete Kac algebra.

Theorem: [Tamatsu 2006] For discrete quantum groups, the firsttwo statements are equivalent. It is open question whether the thirdone is equivalent to the first two.

32

Note: A quantum group G is a discrete if L1(G) is unital. In this case,

its dual quantum group G is compact, i.e., C0(G) is unital. In this case,

we simply write C(G) = C0(G).

It follows from definition that discrete quantum groups are always co-

amenable since L1(G) is unital.

However, compact quantum groups are not necessarily co-amenable.

•Woronowicz’s twisted SUq(2) is co-amenable. So Cu(SUq(2)) = C(SUq(2))

is nuclear.

• Free orthogonal quantum group O+N (N ≥ 3) and free unitary quantum

groups U+N (N ≥ 2) are not co-amenable.

33

Weak Amenability of G

Let G be a locally compact group. For each f ∈ L1(G) = A(G), we candefine a completely bounded map

mf : g ∈ L1(G)→ f g ∈ L1(G).

We say that G is weakly amenable if there exists a net of functions {fi}in L1(G) such that 1) ‖mfi

‖cb ≤ C <∞ for some constant C > 0 and 2)

‖fi ? g − g‖ → 0 for all g ∈ L1(G).

Theorem: [Kraus-R 1999] Let G be a discrete Kac algebra. TFAE:

1. G is weakly amenable,

2. L∞(G) has the weak* CBAP

3. C0(G) = C∗λ(G) has the CBAP.

In this case, we have Λ(G) = Λ(L∞(G)) = Λ(C0(G)).

Remark: For discrete quantum groups, we have 1) ⇒ 2) and 3). Butwe do not know whether 2) or 3) implies 1).

34

Free Quantum Groups [van Dale and Wang 1996]:

Let N ≥ 2 and F ∈ GLN(C) such that FF = R1.

The free orthogonal quantum group O+F is the compact quantum group

given by the universal C*-algebra

Cu(O+F ) = C∗({uij}1≤i,j≤N : U = [uij] is unitary and U = FUF−1}.

If, in addition, F ∈ CUN , then O+F is of Kac type, i.e. the Haar state is

tracial. We use the notion O+N for O+

IN.

The free unitary quantum group U+F is the compact quantum group

given by the universal C*-algebra

Cu(U+F ) = C∗({uij}1≤i,j≤N : U = [uij] and FUF−1}.

Similarly, F is a scalar multiple of unitary if and only if U+F is of Kac

type. We use the notion U+N for U+

IN.

35

Examples:

For N ≥ 3, Brannan first showed (in 2011) that the free othergonalquantum group C(O+

N ) and free unitary quantum group C(O+N ) have

the metric approximation property.

Theorem: [Brannan 2012, Freslon 2013, De Commer-Freslon-

Yamashita 2013] Let N ≥ 2 and F ∈ GLN(C) such that FF = RIN .

The discrete quantum groups FOF = O+F and FUF = U+

F are weaklyamenable with

Λ(FOF ) = Λ(FUF ) = 1.

• The reduced quantum group C*-algebras C(O+F ) and C(U+

F ) have theCCAP.

• The quantum group von Neumann algebras L∞(O+F ) and L∞(U+

F )have the weak* CCAP.

In this case, quantum group C*-algebras C(O+F ) and C(U+

F ) are exact.

36

Haagerup Property

Haagerup has even intensively studied for locally compact groups.

For locally compact quantum groups, we adopt the following (equiva-

lent) definition given by Daws, Fima, Skalski and White in 2013.

Definition: A locally compact quantum group G has the Haagerup

property if the quantum group C*-algebra C0(G) has a bounded ap-

proximate identity {ai} ∈ C0(G) which consists of norm-one positive

definite multipliers/functions.

37

Coefficient Functions of Representations

Let π : L1(G)→ B(H) be a representation of G and let Uπ ∈ L∞(G)⊗B(H)

be the associated unitary operator such that π(ω) = (ω ⊗ ι)(Uπ).

For any ξ, η ∈ H, we call

ϕπξ,η = (ι⊗ ωξ,η)(Uπ) ∈ Cb(G) ⊆ L∞(G)

a coefficient function (associated with the representation).

Remark: For any representation π, there is a unique *-homo π from

Cu(G) = C∗u(G) onto C∗π(G) = π(L1(G))−‖·‖

such that

Uπ = (ι⊗ π)(W).

In this case, we can write

ϕπξ,η = (ι⊗ ωξ,η ◦ π)(W)

and regard it as a coefficient function of µ = ωξ,η ◦ π ∈ Cu(G)∗.

A coefficient is positive definite if it has the form ϕπξ,ξ.

38

Von Neumann Algebra Haagerup Property

Haagerup property for finite von Neumann algebras (with trace) was

first studied by Connes in 1985. A definition for general von Neumann

algebras is given recently (both in 2013) by two independent groups:

Caspers-Skalski and Okayasu-Tomatsu.

Definition: A von Neumann algebra M with a normal faithful weight

ϕ has the Haagerup property if there exists a net of unital normal cp

maps Φi on M such that

0) ϕ ◦Φi ≤ ϕ1) each Φi extends to a compact operator on L2(M,ϕ)

2) ‖Φi(x)− x‖2 → 0 for every x ∈M (resp. for every x ∈ L2(M,ϕ)).

39

Theorem: [Choda 1983] A discrete group G has the Haagerup prop-

erty if and only if its group von Neumann algebra L∞(G) = L(G) (with

the canonical trace) has the von Neumann algebra Haagerup property.

Theorem: [Daws, Fima, Skalski, White 2013] Let G be a discrete

Kac algebra. Then G has the Haagerup property if and only if L∞(G)

(with the canonical trace) has the von Neumann algebra Haagerup prop-

erty.

40

Examples:

• Every co-amenable discrete quantum groups has the Haagerup prop-

erty.

Theorem: [Brannan 2012, De Commer-Freslon-Yamashita 2013]

The discrete quantum groups FOF and FUF have the Haagerup property.

41

Exotic Quantum Group C*-algebras

42

Exotic C*-algebras

Let us recall that a quantum group G is co-amenable if and only if

Cu(G) = C0(G).

So if G is not co-amenable, then we have a proper quotient

Cu(G) 7→ C0(G)

Question: If G is not co-amenable, do we have any (quantum group)

C*-algebra A, sitting in between Cu(G) and C0(G), i.e., do we have any

(quantum group) C*-algebra A such that we get proper quotients

Cu(G) 7→ A 7→ C0(G).

We call such C*-algebra A an exotic C*-algebra.

43

Brown-Guentner’s D- C*-algebras C∗D(G)

Let G be a discrete group and D an ideal in `∞(G). A unitary repre-

sentation π : G → B(H) is called an D-representation if there eixsts a

dense subset H0 ⊆ H such that the coefficient functions

πξ,η(s) = 〈π(s)ξ|η〉 ∈ D

for all ξ, η ∈ H0.

• Direct sum of D-representations is a D-representation

• Tensor product of a D-representations is a D-representation.

• If Cc(G) ⊆ D ⊆ `∞(G), then for any f ∈ `1(G),

‖f‖D = {‖π(f)‖ : π is a D-representation}

defines a faithful C*-algebra norm on `1(G). The associated C*-algebra

C∗D(G) is a compact quantum group C*-algebra.

44

Quantum Group Approach to D- C*-algebras C∗D(G)

This is a recent joint work with Michael Brannan.

Let us assume that G is a unimodular discrete quantum group. In this

case, we get

`∞(G) =∏α∈I B(Hα) with tracial Haar weight h = ⊕α∈IdαTrα

C0(G) = c0 −⊕α∈I B(Hα)

Cc(G) =⊕α∈I B(Hα).

`1(G) = `1 −⊕α∈I B(Hα)∗

`p(G) = `p −⊕α∈I Lp(Hα)

45

Let D a subspace in `∞(G). A representation

π : ω ∈ `1(G)→ π(ω) = (ω ⊗ ι)(Uπ) ∈ B(H)

is an D-representation if there eixsts a dense subset H0 ⊆ H such that

the coefficient functions

ϕπξ,η = (ι⊗ ωξ,η)(Uπ) ∈ D

for all ξ, η ∈ H0.

• Direct sum of D-representations is a D-representation

• Tensor product of a D-representations is a D-representation.

• If Cc(G) ⊆ D ⊆ `∞(G), then for any ω ∈ `1(G),

‖ω‖D = {‖π(ω)‖ : π is a D-representation}

defines a faithful C*-algebra norm on `1(G). The associated C*-algebra

C∗D(G) is a compact quantum group C*-algebra.

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It is clear that if Cc(G) ⊆ D1 ⊆ D2 ⊆ `∞(G), then we have the quotient

C∗D2→ C∗D1

.

It is also easy to see that C∗`∞(G) = C∗(G).

Theorem: [Brown-Guenter 2012, Brannan-R 2014]

1) If Cc(G) ⊆ D ⊆ `2(G), then C∗D(G) = C∗λ(G).

2) G is amenable if and only if there exists an 1 ≤ p <∞ such that

C∗`p(G) = C∗u(G).

3) G has the Haagerup property if and only if C∗(G) = C∗C0(G)(G).

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Theorem: [Brown-Guentner 2012] For d ≥ 2, there exists some

2 < p <∞ such that C∗`p(Fd) is exotic, i.e., the quotents

C∗(Fd)→ C∗`p(Fd)→ C∗λ(Fd)

are proper.

Theorem : [Okayasu 2012] Let 2 < p < q <∞. The quotients

C∗(Fd)→ C∗`q(Fd)→ C∗`p(Fd)→ C∗λ(Fd)

are all proper. So there are uncountablly many of exotic compact quan-

tum group C*-algebras C∗`p(Fd).

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Quantum Group Case

Let N ≥ 3 and F ∈ GLN(C) such that FF ∈ RIN and F ∈ CUN . In this

case, FOF and FUF are unimodular discrete quantum groups.

Theorem : [Brannan-R 2014] Let 2 < p < q <∞. The quotients

C∗(FOF )→ C∗`q(FOF )→ C∗`p(FOF )→ C∗λ(FOF ) = C(O+F )

and

C∗(FUF )→ C∗`q(FUF )→ C∗`p(FUF )→ C∗λ(FUF ) = C(U+F )

are all proper. So we get uncountablly many of exotic compact quantum

group C*-algebras.

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The main ingredient in the proof for the FOF case is that we need

• Haagerup property

• Rapid decay property

More precisely, we consider a family of positive definite functions

ϕr =∏

n∈N0=Irr(O+F )

Sn(rN)

Sn(N)1B(Hn) ∈ C0(FOF ) (0 < 1 < r < 1)

where (Sn) are type 2 Chebyschev polynomials defined by

S0(x) = 1, S1(x) = 1, and Sn(x) = xSn(x)− Sn−1(x) (n ≥ 1).

Since each ϕr is a positive definite function in C0(FOF ), it corresponds

a positive linear functional µr ∈ C∗(FOF )∗ = B(FOF ).

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Sinse 2 < p < q <∞, we have the natural quotients

C∗(FOF )→ C∗`q(FOF )→ C∗`p(FOF )→ C∗λ(FOF ) = C(O+F )

and thus get the inclusions

C∗(FOF )∗ ← C∗`q(FOF )∗ ← C∗`p(FOF )∗ ← C∗λ(FOF )∗ = C(O+F )∗.

Then we can show that if 2 < p < q <∞, then there exists r0 such that

ϕr0 corresponds to a positive linear functional in C∗`q(FOF ), but not in

C∗`p(FOF ). This completes the proof.

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Thank you !

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