Unitary spherical representations of Drinfeld doubles

Definition A.1.

We say G^ has central property (T) if the following holds:

If a net of central states (ωi)⊂Cu⁢(G)* converges to ε in the weak*-topology, then it converges in norm.

Remark A.2.

According to [14, Theorem 3.1], property (T) in the sense of [8] is equivalent to the following:

If a net of states (ωi) converges to ε in the weak*-topology, then it converges in norm.

In particular, property (T) implies central property (T).

Lemma A.3.

Let A be a C*-algebra and χ a *-character of A. Then the following are equivalent.

1. The character χ is isolated in the unitary dual of A.

2. If a net of states (ωi) converges to χ in the weak*-topology, then it converges in norm.

3. There exists a central projection pχ in A such that for any a∈A,

   a⁢pχ=χ⁢(a)⁢pχ.

Proof.

The equivalence (i) ⇔ (ii) is immediate from the definition of the topology on the unitary dual. For the implication (ii) ⇒ (iii) see [3, Theorem 17.2.4]. (The proof works for general C*-algebras.)

We show (iii) ⇒ (i). Applying [6, Proposition 3.2.1] to A=pχ⁢A⊕(1-pχ)⁢A, we can decompose the unitary dual of A into a disjoint union of those of pχ⁢A and (1-pχ)⁢A. Since the unitary dual of pχ⁢A is {χ}, χ is isolated. ∎

Proposition A.4.

The following are equivalent.

1. The quantum group G^ has central property (T).

2. If a net of central completely positive multipliers (Tiu) on Cu⁢(G) converges to the identity pointwisely, then it converges in norm.

Proposition A.5.

Let G and H be compact quantum groups which are monoidally equivalent. Then we have a one-to-one correspondence between central completely bounded multipliers on Cu⁢(G) and Cu⁢(H) preserving the norm and the weak*-topology.

In particular, if G^ has central property (T), so does H^.

Lemma A.6.

Let G be a compact quantum group of Kac type. Then there exists a conditional expectation

such that

Proof.

Let us consider the left and right regular representations:

Notice that

where Ω is the canonical cyclic vector in L2⁢(G).

First, we claim

is injective. Using Fell’s absorption, we get that

is a well-defined representation. Now

shows α is injective.

Finally, consider the φ⊗φ-preserving conditional expectation

Then the image of (EΔ⊗id)⁢α is α⁢(Cu⁢(Char⁢G)) and

is the desired conditional expectation. ∎

Proposition A.7.

Let G be a compact quantum group of Kac type. Then the following are equivalent.

1. The quantum group G^ has property (T).

2. The quantum group G^ has central property (T).

3. If a net of states (ωi) on the character algebra Cu⁢(Char⁢(G)) converges to ε in the weak*-topology, it converges in norm.

Proof.

As we observed, (i) implies (ii).

To show that (ii) implies (iii), suppose G^ has property (T). Take a net of states (ωi) on Cu⁢(Char⁢(G)) converging to ε in the weak*-topology. By Lemma A.6, there exists a conditional expectation

Then (ωi∘E) is a net of central states on Cu⁢(G) converging to ε in the weak*-topology. Hence it converges in norm. Since ωi∘E is nothing but ωi on Cu⁢(Char⁢(G)), the net (ωi) converges to ε in norm.

To show that (iii) implies (i), take the projection pε∈Cu⁢(Char⁢(G)) as in Lemma A.3. We claim x⁢pε=pε⁢x=ε⁢(x)⁢pε. To see this, embed Cu⁢(G) into B⁢(H) for a Hilbert space H. Then

Since ui⁢iπ is a contraction, this shows

In a similar way,

Since ∑k(uk⁢iπ)*⁢uk⁢jπ=1, we have

that is,

Taking * of the formula, we get pε is central. Again from Lemma A.3, G^ has property (T). ∎

Proposition A.8.

Suppose the Drinfeld double D⁢(G) has property (T). Then G^ has central property (T).

Proof.

Thanks to Lemma A.3, the assumption is equivalent to the following: If a net of states on 𝒪⁢(G)⋈cc⁢(G^) converges to the counit εD, then it converges in norm.

Take central states ω,μ on Cu⁢(G). Let π:Cu⁢(G)→M⁢(C0u⁢(D^⁢(G))) be the *-homomorphism extending 𝒪(G)→M(𝒪(G)⋈cc(G^)). Then since ω=Ind⁡ω∘π, we have

In particular, for a net of central states (ωi), if (Ind⁡ωi) converges to Ind⁡ω in norm, then (ωi) converges to ω in norm.

Hence if we have a net of central states (ωi) converging to ε in the weak*-topology, then (Ind⁡ωi) converges to εD=Ind⁡ε in the weak*-topology. By assumption, it converges in norm, which turns out to be the convergence of (ωi) to ε in norm, which is desired. ∎

Corollary A.9.

The discrete quantum group SUq⁢(2⁢n+1)^ has central property (T).

Proposition A.10.

Let F:Rep⁢(SUq⁢(3))→Hilbf be a unitary fiber functor. Let x∈Rep⁢(SUq⁢(3)) be the fundamental representation of SUq⁢(3). Then we have either

Proof.

In [5], it is shown that for any compact quantum group G,

is a central state.

For a unitary fiber functor F, take the corresponding quantum group G which is monoidally equivalent to SUq⁢(3). Then by Proposition A.5,

is a central state on SUq⁢(3). On the other hand, the central state corresponding to ν∈X is nothing but

Put N:=dim⁡F⁢(x) and find t≥0 such that

Then ωt⁢ρ=ωF. Therefore it is positive definite if and only if t⁢ρ is in the list in Theorem 7.4, hence t≤1 or t=2. ∎