Unitarizable Groups and Quantum Groupsbanach2019/pdf/Brannan.pdf · Unitarizable Groups and Quantum Groups Michael Brannan Texas A&M University Banach Algebras and Applications, July

Unitarizable Groups and Quantum Groups

Michael Brannan

Texas A&M University

Banach Algebras and Applications, July 12, 2019

Unitarizable groups

Let G be a locally compact group, H an Hilbert space.

A (uniformly bounded) representation of G on H is a grouphomomorphism π : G→ B(H)inv, continous with respect to theSOT and uniformly bounded: ‖π‖ := supt∈G ‖π(t)‖ <∞.

We call π a unitary representation if π(G) ⊂ U(H), the unitarygroup of H.

We say π is unitarizable if there exists T ∈ B(H)inv so thatTπ(·)T−1 is a unitary representation.

A locally compact group G is unitarizable if every representation ofG is unitarizable.

QuestionWhich locally compact groups are unitarizable?

Basic Examples: Finite and compact groups are unitarizable. Anygroup containing a closed free subgroup is not unitarizable.

Unitarizable groups








Unitarizable groups








Unitarizable groups








Amenability and Unitarizability

Theorem (Day-Dixmier 1950)

Amenable groups are always unitarizable.

Recall: G is amenable if it has a (left) translation-invariant mean.I.e., there exists a state m : L∞(G)→ C such that∫

Gf(t)dm(t) =

∫Gf(st)dm(t) (f ∈ L∞(G), s ∈ G).

Sketch.Given a representation π : G→ B(H)inv, let

S =

∫Gπ(t)∗π(t)dm(t) (m : L∞(G)→ C an invariant mean),

and put T =√S ∈ B(H)inv. Then Tπ(·)T−1 is unitary (and

moreover we have ‖T‖‖T−1‖ ≤ ‖π‖2).

Amenability and Unitarizability

Theorem (Day-Dixmier 1950)

Amenable groups are always unitarizable.

Recall: G is amenable if it has a (left) translation-invariant mean.I.e., there exists a state m : L∞(G)→ C such that∫

Gf(t)dm(t) =

∫Gf(st)dm(t) (f ∈ L∞(G), s ∈ G).

Sketch.Given a representation π : G→ B(H)inv, let

S =

∫Gπ(t)∗π(t)dm(t) (m : L∞(G)→ C an invariant mean),

and put T =√S ∈ B(H)inv. Then Tπ(·)T−1 is unitary (and

moreover we have ‖T‖‖T−1‖ ≤ ‖π‖2).

Dixmier’s Similarity Problem for Groups

We know that G amenable =⇒ G unitarizable. Is the conversetrue?

Question (Dixmier’s Similarity Problem for Groups ∼1950)

Is every unitarizable group amenable?

I This problem is still open!

I However, significant progress has been made by Pisier (first fordiscrete groups) and then Spronk (locally compact groups).

Theorem (Pisier ∼’98, Spronk ’02)

If G is unitarizable, then there exist constants K, γ > 0 such thatfor any representation π : G→ B(H), ∃T ∈ B(H)inv such that

Tπ(·)T−1 is unitary and ‖T‖‖T−1‖ ≤ K‖π‖γ .

Moreover, if γ < 3, then G is amenable.





















More Recent Work on Dixmier’s Problem

More recently, Monod and Ozawa have shown:

Theorem (Monod-Ozawa ’10)

For a discrete group G, the following are equivalent:

1. G is amenable

2. For any abelian group A, the lamplighter groupG oA :=

⊕GAoG is unitarizable.

Beyond the above mentioned works, no new progress has beenmade on the Dixmier problem for groups!

Our goal today: Instead, we switch gears and study connectionsbetween unitarizability and amenability for more general objects,called quantum groups.

More Recent Work on Dixmier’s Problem

More recently, Monod and Ozawa have shown:

Theorem (Monod-Ozawa ’10)

For a discrete group G, the following are equivalent:

1. G is amenable

2. For any abelian group A, the lamplighter groupG oA :=

⊕GAoG is unitarizable.

Beyond the above mentioned works, no new progress has beenmade on the Dixmier problem for groups!

Our goal today: Instead, we switch gears and study connectionsbetween unitarizability and amenability for more general objects,called quantum groups.

From Groups to Quantum Groups - MotivationGiven a locally compact group G, we can fully encode its structurethrough various operator algebraic structures associated to G.

For example, given G, we can associate the following structures:I The (abelian) von Neumann algebra M := L∞(G).I The normal unital ∗-homomorphism

∆ : M →M⊗M ; ∆f(s, t) = f(st),

which satisfies the co-associativity identity

(ι⊗∆)∆ = (∆⊗ ι)∆.I Two (normal, semi-finite) Haar weights

ϕL, ϕR : M → C,

given by integration with respect to left/right Haar measure.They satisfy the left/right invariance identities

(ι⊗ ϕL)∆ = ϕL(·)1 & (ϕR ⊗ ι)∆ = ϕR(·)1.Fact: This quadruple of (M,∆, ϕL, ϕR) completely determines Gup to isomorphism.



∆ : M →M⊗M ; ∆f(s, t) = f(st),



ϕL, ϕR : M → C,


(ι⊗ ϕL)∆ = ϕL(·)1 & (ϕR ⊗ ι)∆ = ϕR(·)1.

Fact: This quadruple of (M,∆, ϕL, ϕR) completely determines Gup to isomorphism.



∆ : M →M⊗M ; ∆f(s, t) = f(st),



ϕL, ϕR : M → C,


(ι⊗ ϕL)∆ = ϕL(·)1 & (ϕR ⊗ ι)∆ = ϕR(·)1.Fact: This quadruple of (M,∆, ϕL, ϕR) completely determines Gup to isomorphism.

Locally Compact Quantum Groups

A locally compact quantum group is simply a non-commutativegeneralization of the algebraic structures we just associated to G:

Definition (Kustermans-Vaes 2000)

A locally compact quantum group is a quadrupleG := (M,∆, ϕL, ϕR), where

I M is a von Neumann algebra.

I ∆ : M →M⊗M is a normal unital ∗-homomorphismsatisfying co-associativity: (ι⊗∆)∆ = (∆⊗ ι)∆).

I ϕL, ϕR : M → C are normal semifinite weights satisfying theleft/right invariance conditions

(ι⊗ ϕL)∆ = ϕL(·)1 & (ϕR ⊗ ι)∆ = ϕR(·)1.

For any locally compact quantum group (LCQG) G, we writeM = L∞(G).

Locally Compact Quantum Groups

A locally compact quantum group is simply a non-commutativegeneralization of the algebraic structures we just associated to G:

Definition (Kustermans-Vaes 2000)

A locally compact quantum group is a quadrupleG := (M,∆, ϕL, ϕR), where

I M is a von Neumann algebra.

I ∆ : M →M⊗M is a normal unital ∗-homomorphismsatisfying co-associativity: (ι⊗∆)∆ = (∆⊗ ι)∆).

I ϕL, ϕR : M → C are normal semifinite weights satisfying theleft/right invariance conditions

(ι⊗ ϕL)∆ = ϕL(·)1 & (ϕR ⊗ ι)∆ = ϕR(·)1.

For any locally compact quantum group (LCQG) G, we writeM = L∞(G).

Basic Examples

Let G := (L∞(G),∆, ϕL, ϕR).

Classical locally compact groups G: Those LCQG G withL∞(G) = L∞(G), ∆, ϕL, ϕR constructed from G, as before.Conversely, any LCQG G with abelian L∞(G) arises in this way.

The Pontryagin dual of a locally compact group G:- Notation: G = G.- Let λ : G→ U(L2(G)) be the left-regular representation.- Put L∞(G) := V N(G) = λ(G)′′ ⊂ B(L2(G)).- ∆(λ(t)) = λ(t)⊗ λ(t).- ϕL = ϕR = Plancherel weight:

ϕL(λ(f)) = f(e) (f ∈ Cc(G)).

ThenG = (V N(G),∆, ϕL, ϕR)

is the LCQG dual to G.

Basic Examples

Let G := (L∞(G),∆, ϕL, ϕR).

Classical locally compact groups G: Those LCQG G withL∞(G) = L∞(G), ∆, ϕL, ϕR constructed from G, as before.Conversely, any LCQG G with abelian L∞(G) arises in this way.

The Pontryagin dual of a locally compact group G:- Notation: G = G.- Let λ : G→ U(L2(G)) be the left-regular representation.- Put L∞(G) := V N(G) = λ(G)′′ ⊂ B(L2(G)).- ∆(λ(t)) = λ(t)⊗ λ(t).- ϕL = ϕR = Plancherel weight:

ϕL(λ(f)) = f(e) (f ∈ Cc(G)).

ThenG = (V N(G),∆, ϕL, ϕR)

is the LCQG dual to G.

Compact Matrix Quantum GroupsGenuinely quantum examples can be “easily” constructed asfollows: Let A be a unital ∗-algebra and u = [uij ] ∈Mn(A).Assume thatI A is generated as a ∗-algebra by the entries uij of u.I u is unitary in Mn(A): u∗u = uu∗ = 1Mn(A).I The conjugate matrix u := [u∗ij ] is invertible in Mn(A).

Theorem (Woronowicz ’87)

If (A, u) are as above, then A admits a state h : A → C that“completes” A to a LCQG G with

L∞(G) = πh(A)′′ ⊂ B(L2(A, h)),

∆(uij) =∑k

uik ⊗ ukj , & ϕL = ϕR = h.

We call such G a Compact matrix quantum group. HereA = A(G) plays the role of “non-commutative algebra ofpolynomial functions” on G.

More Examples

1. q-deformed SU(2) quantum group: Given q ∈ [−1, 1]\{0},the quantum group SUq(2) is given by:

A(SUq(2)) = ∗-alg(α, γ

∣∣∣u =

[α −qγ∗γ α∗

]is unitary

).

Note: At q = 1, SU1(2) = SU(2).Note: When q 6= ±1, the Haar state h is non-tracial

2. Quantum permutation group S+n :

A(S+n ) = ∗-alg

({uij}ni,j=1

∣∣∣u = [uij ] is unitary & u2ij = uij = u∗ij

).

Note: Haar state h is tracial for all n.

More Examples

1. q-deformed SU(2) quantum group: Given q ∈ [−1, 1]\{0},the quantum group SUq(2) is given by:

A(SUq(2)) = ∗-alg(α, γ

∣∣∣u =

[α −qγ∗γ α∗

]is unitary

).

Note: At q = 1, SU1(2) = SU(2).Note: When q 6= ±1, the Haar state h is non-tracial

2. Quantum permutation group S+n :

A(S+n ) = ∗-alg

({uij}ni,j=1

∣∣∣u = [uij ] is unitary & u2ij = uij = u∗ij

).

Note: Haar state h is tracial for all n.

Compactness and Amenability for Quantum Groups

I A LCQG G is compact if the Haar weights ϕL, ϕR can betaken as states. Compact matrix quantum groups like SUq(2)and S+

n are all compact.

I A LCQG is called amenable if there exists a (left) invariantmean on L∞(G). I.e., a state m : L∞(G)→ C satisfying

(ι⊗m)∆ = m(·)1.

I All compact quantum groups are amenable.

I The Pontryagin dual of group G is amenable: Just takem : V N(G)→ C a weak-∗ cluster point of any net (uα)α ofpositive definite functions satisfying

uα(e) = 1 & suppuα → {e}.

Representations of Quantum Groups

How do we define representations of LCQGs?

For a locally compact group G, we have a bijective correspondencebetween representations

π : G→ B(H)inv

and non-degenerate completely bounded homomorphisms

π1 : L1(G)→ B(H), with π1(f) =

∫Gf(t)π(t)dt.

Moreover, we have

I ‖π1‖cb = ‖π1‖ = ‖π‖.I π unitary ⇐⇒ π1 is a ∗-homomorphism.

I π is unitarizable ⇐⇒ π1 is similar to a ∗-homomorphism.

I G is unitarizable ⇐⇒ every c.b. homo. π1 : L1(G)→ B(H)is similar to a ∗-homomorphism.


How do we define representations of LCQGs?For a locally compact group G, we have a bijective correspondencebetween representations

π : G→ B(H)inv

and non-degenerate completely bounded homomorphisms

π1 : L1(G)→ B(H), with π1(f) =

∫Gf(t)π(t)dt.

Moreover, we have

I ‖π1‖cb = ‖π1‖ = ‖π‖.I π unitary ⇐⇒ π1 is a ∗-homomorphism.

I π is unitarizable ⇐⇒ π1 is similar to a ∗-homomorphism.

I G is unitarizable ⇐⇒ every c.b. homo. π1 : L1(G)→ B(H)is similar to a ∗-homomorphism.


For a quantum group G, we have an analogue of L1(G): Define

L1(G) := (L∞(G))∗,

and define a multiplication ? on L1(G) via

f1 ? f2 := (f1 ⊗ f2)∆ (fi ∈ L1(G)).

Then L1(G) becomes a completely contractive Banach algebra,called the convolution algebra of G.

Fact: L1(G) always admits a canonical (densely defined)involution ] : L1(G)→ L1(G), generalizing the classical case.

We define a representation of G on a Hilbert space H to be anon-degenerate completely bounded homomorphism

π : L1(G)→ B(H).



L1(G) := (L∞(G))∗,


f1 ? f2 := (f1 ⊗ f2)∆ (fi ∈ L1(G)).




π : L1(G)→ B(H).



L1(G) := (L∞(G))∗,


f1 ? f2 := (f1 ⊗ f2)∆ (fi ∈ L1(G)).




π : L1(G)→ B(H).

Unitary Representations

Using the involution ] on L1(G), we can talk about∗-homomorphisms π : L1(G)→ B(H). We call thesehomomorphisms unitary representations of G.

Why call them “unitary”???: By the operator space duality,

CB(L1(G), B(H)) ∼= (L1(G)⊗T (H))∗ ∼= L∞(G)⊗B(H),

so any cb homomorphism π : L1(G)→ B(H) corresponds to acorepresentation operator

Vπ ∈ L∞(G)⊗B(H); π(f) = (f ⊗ ι)Vπ (f ∈ L1(G))

and we have by [Kustermanns]

π is a ∗-homomorphism ⇐⇒ Vπ is unitary.

Unitary Representations

Using the involution ] on L1(G), we can talk about∗-homomorphisms π : L1(G)→ B(H). We call thesehomomorphisms unitary representations of G.

Why call them “unitary”???: By the operator space duality,

CB(L1(G), B(H)) ∼= (L1(G)⊗T (H))∗ ∼= L∞(G)⊗B(H),

so any cb homomorphism π : L1(G)→ B(H) corresponds to acorepresentation operator

Vπ ∈ L∞(G)⊗B(H); π(f) = (f ⊗ ι)Vπ (f ∈ L1(G))

and we have by [Kustermanns]

π is a ∗-homomorphism ⇐⇒ Vπ is unitary.

Key Example: Fourier Algebras

Consider the dual G of a locally compact group G. Then

I L1(G) = V N(G)∗ = A(G), the Fourier algebra of G.

A(G) = {t 7→ uξ,η(t) = 〈λ(t)ξ|η〉∣∣ξ, η ∈ L2(G)} ⊂ C0(G).

is the space of coefficient functions of the left regularrepresentation. It is a commutative, regular, completelycontractive Banach ∗-algebra in C0(G) with spectrum G andnorm

‖u‖A(G) = inf{‖ξ‖‖η‖ : u = uξ,η

}.

I Representations of G are completely bounded homomorphismsπ : A(G)→ B(H).

I Unitary representations of G are ∗-homomorphismsπ : A(G)→ B(H). They arise uniquely (by restriction) from∗-homomorphisms π : C0(G)→ B(H).

Key Example: Fourier Algebras

Consider the dual G of a locally compact group G. Then

I L1(G) = V N(G)∗ = A(G), the Fourier algebra of G.

A(G) = {t 7→ uξ,η(t) = 〈λ(t)ξ|η〉∣∣ξ, η ∈ L2(G)} ⊂ C0(G).

is the space of coefficient functions of the left regularrepresentation. It is a commutative, regular, completelycontractive Banach ∗-algebra in C0(G) with spectrum G andnorm

‖u‖A(G) = inf{‖ξ‖‖η‖ : u = uξ,η

}.

I Representations of G are completely bounded homomorphismsπ : A(G)→ B(H).

I Unitary representations of G are ∗-homomorphismsπ : A(G)→ B(H). They arise uniquely (by restriction) from∗-homomorphisms π : C0(G)→ B(H).

Unitarizable Quantum Groups

In general we call a LCQG G unitarizable if every cbhomomorphism π : L1(G)→ B(H) is similar to a∗-homomorphism.

Equivalently, G is unitarizable iff every corepresentation operatorVπ ∈ L∞(G)⊗B(H) admits T ∈ B(H)inv so that

(ι⊗ T )Vπ(ι⊗ T−1) is unitary.

QuestionWhich quantum groups are unitarizable?

In particular, is there a quantum version of the Day-Dixmiertheorem (saying amenablility =⇒ unitarizability)?

Partial Results: The Fourier Algebras

Recall that the dual G of a locally compact group is alwaysamenable.

Conjecture (Effros-Ruan, Spronk)

For any locally compact group G, G is unitarizable.

I That is, every cb homomorphism π : A(G)→ B(H) is similarto a ∗-homomorphism.

I Moreover, the “Similarity Degree” of A(G) equals 2: That is,given π : A(G)→ B(H), we can always find a similarityT ∈ B(H)inv making

Tπ(·)T−1 a ∗-homomorphism with ‖T‖‖T−1‖ ≤ ‖π‖2cb.

Note: A cb homomorphism π : A(G)→ B(H) will be similar to a∗-homomorphism iff it extends to a bounded homomorphism fromC0(G)→ B(H).


Recall that the dual G of a locally compact group is alwaysamenable.

Conjecture (Effros-Ruan, Spronk)

For any locally compact group G, G is unitarizable.

I That is, every cb homomorphism π : A(G)→ B(H) is similarto a ∗-homomorphism.

I Moreover, the “Similarity Degree” of A(G) equals 2: That is,given π : A(G)→ B(H), we can always find a similarityT ∈ B(H)inv making

Tπ(·)T−1 a ∗-homomorphism with ‖T‖‖T−1‖ ≤ ‖π‖2cb.

Note: A cb homomorphism π : A(G)→ B(H) will be similar to a∗-homomorphism iff it extends to a bounded homomorphism fromC0(G)→ B(H).


Theorem (B.-Samei ’10, B.-Daws-Samei ’13)

Let G be a [SIN]-group (e.g., compact/discrete) or an amenablegroup with an open [SIN]-subgroup. Then G is unitarizable.Moreover, A(G) has similarity degree ≤ 4: Givenπ : A(G)→ B(H), we can always find a similarity S ∈ B(H)invmaking

Sπ(·)S−1 a ∗-homomorphism with ‖S‖‖S−1‖ ≤ ‖π‖4cb.

Current State of the Art: Building on Pisier’s work on thesimilarity problem, Lee, Samei and Spronk obtained:

Theorem (Lee-Samei-Spronk ’16)

If G is a quasi-SIN group (e.g., amenable, compact, discrete, SIN),then G is unitarizable.


Theorem (B.-Samei ’10, B.-Daws-Samei ’13)

Let G be a [SIN]-group (e.g., compact/discrete) or an amenablegroup with an open [SIN]-subgroup. Then G is unitarizable.Moreover, A(G) has similarity degree ≤ 4: Givenπ : A(G)→ B(H), we can always find a similarity S ∈ B(H)invmaking

Sπ(·)S−1 a ∗-homomorphism with ‖S‖‖S−1‖ ≤ ‖π‖4cb.

Current State of the Art: Building on Pisier’s work on thesimilarity problem, Lee, Samei and Spronk obtained:

Theorem (Lee-Samei-Spronk ’16)

If G is a quasi-SIN group (e.g., amenable, compact, discrete, SIN),then G is unitarizable.

Aside: Are cb Homomorphism Really Necessary?

QuestionFor A(G), why not just consider bounded homomorphisms?

Answer: Any homo. π : A(G)→ B(H) that is similar to a∗-homo. must be completely bounded! So we must restrict to cbmaps.

QuestionDo there exist bounded, but not completely bounded,homomorphisms π : A(G)→ B(H)?

Answer: Yes!

Theorem (Choi-Samei ’13)

Let G contain F2 as a closed subgroup. Then there exist a boundedrepresentation π : A(G)→ B(H) that is not completely bounded.

Aside: Are cb Homomorphism Really Necessary?

QuestionFor A(G), why not just consider bounded homomorphisms?

Answer: Any homo. π : A(G)→ B(H) that is similar to a∗-homo. must be completely bounded! So we must restrict to cbmaps.

QuestionDo there exist bounded, but not completely bounded,homomorphisms π : A(G)→ B(H)?

Answer: Yes!

Theorem (Choi-Samei ’13)

Let G contain F2 as a closed subgroup. Then there exist a boundedrepresentation π : A(G)→ B(H) that is not completely bounded.

Partial Results: For Compact Quantum GroupsLet G be a compact quantum group (in particular, G is amenable).

QuestionIs G unitarizable? (Is every cb homo π : L1(G)→ B(H) similar toa ∗-homo?)

Theorem (Woronowicz ’90’s)

Every finite-dimensional representation of a compact quantumgroup is unitarizable.

Theorem (B-Daws-Samei ’13)

Every compact quantum group with tracial Haar state isunitarizable. (E.g., S+

n is unitarizable).

QuestionWhat about G = SUq(2) (q 6= ±1)? Is it unitarizable?

AnswerNo.







n is unitarizable).


AnswerNo.







n is unitarizable).


AnswerNo.







n is unitarizable).


AnswerNo.

SUq(2) is not unitarizable

This is joint work with Sang-Gyun Youn, Queen’s University.

Theorem (B.-Youn ’18)

There exists a cb homomorphism π : L1(SUq(2))→ B(H) that isnot similar to ∗-homomorphism.

More generally, we can prove:

Theorem (B.-Youn.)

Let G be a compact simply connected semisimple Lie group and letGq (0 < q < 1) be the Drinfeld-Jimbo q-deformation of G. Thenthe compact quantum group Gq is non-unitarizable.

RemarkEven for SUq(2), our arguments are non-constructive and do notsupply explicit examples of non-unitarizable representations.

ProblemConstruct explicit examples!

Sketch of the Proof - for SUq(2)

The key point here is to exploit a connection between

1. the modular theory of the Haar state ϕ on SUq(2),

2. and Pisier’s work on similarity degree of completelycontractive Banach algebras.

Our arguments are similar in spirit to [Caspers-Lee-Ricard]’s recentwork on the failure of operator biprojectivity for L1(SUq(2)).

So, let’s assume SUq(2) is unitarizable. It then follows from Pisierthat L1(SUq(2)) has finite completely bounded similarity degree:

There exist universal constants K, γ > 0 such that for everycompletely bounded homomorphism π : L1(SUq(2))→ B(H),∃T ∈ B(H)inv so that

I Tπ(·)T−1 is a ∗-representation.

I ‖T‖‖T−1‖ ≤ K‖π‖γcb.

Sketch of the Proof - for SUq(2)

The key point here is to exploit a connection between

1. the modular theory of the Haar state ϕ on SUq(2),

2. and Pisier’s work on similarity degree of completelycontractive Banach algebras.

Our arguments are similar in spirit to [Caspers-Lee-Ricard]’s recentwork on the failure of operator biprojectivity for L1(SUq(2)).

So, let’s assume SUq(2) is unitarizable. It then follows from Pisierthat L1(SUq(2)) has finite completely bounded similarity degree:

There exist universal constants K, γ > 0 such that for everycompletely bounded homomorphism π : L1(SUq(2))→ B(H),∃T ∈ B(H)inv so that

I Tπ(·)T−1 is a ∗-representation.

I ‖T‖‖T−1‖ ≤ K‖π‖γcb.

Sketch of the proof, ctdNext we consider some facts about the irreducible representationsof SUq(2) (due to Woronowicz):

For each s ∈ N0, ∃! irreducible ∗-representationπs : L1(SUq(2))→Ms+1(C), with unitary corepresentationoperator Vπs = [vsij ] ∈Ms+1(L∞(SUq(2))).

For each s, consider the contragradient representation πs of πs.It’s (non-unitary) corepresentation is Vπs = [(vsij)

∗].

Note that ‖πs‖cb = ‖Vπs‖ ≤ (s+ 1)2.

Woronowicz showed that πs is similar to πs, with similarity Tssatisfying

‖Ts‖ = ‖T−1s ‖ = |q|−s/2.

On the other hand, we already know that πs is also similar to πsvia a similarity Ts satisfying

‖Ts‖‖T−1s ‖ ≤ K‖πs‖

γcb ≤ K(s+ 1)2γ .




∗].



‖Ts‖ = ‖T−1s ‖ = |q|−s/2.


‖Ts‖‖T−1s ‖ ≤ K‖πs‖

γcb ≤ K(s+ 1)2γ .




∗].



‖Ts‖ = ‖T−1s ‖ = |q|−s/2.


‖Ts‖‖T−1s ‖ ≤ K‖πs‖

γcb ≤ K(s+ 1)2γ .




∗].



‖Ts‖ = ‖T−1s ‖ = |q|−s/2.


‖Ts‖‖T−1s ‖ ≤ K‖πs‖

γcb ≤ K(s+ 1)2γ .




∗].



‖Ts‖ = ‖T−1s ‖ = |q|−s/2.


‖Ts‖‖T−1s ‖ ≤ K‖πs‖

γcb ≤ K(s+ 1)2γ .

Sketch of the proof, ctd

But, πs is irreducible, so Schur’s lemma implies that ∃ λ ∈ C sothat

Ts = λTs

In particular,

|q|−s = ‖Ts‖‖T−1s ‖ = ‖Ts‖‖T−1

s ‖ ≤ K(s+ 1)2γ (∀s ∈ N0).

Contradiction!

Unitarizable Groups and Quantum Groupsbanach2019/pdf/Brannan.pdf · Unitarizable Groups and Quantum Groups Michael Brannan Texas A&M University Banach Algebras and Applications, July

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