Inductive limits of compact quantum groups and their unitary representations

Abstract

We introduce inductive limits of compact quantum groups and their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantization of characters and central probability measures in the asymptotic representation theory. We also study tensor product representations of the infinite-dimensional quantum unitary group. As a by-product, we give an explicit representation-theoretic interpretation to certain transformations that play an important role in analyzing q-central probability measures on the paths in the q-Gelfand–Tsetlin graph.

Appendix A. Spherical representations and spherical functions

This appendix is based on what Yoshimichi Ueda explained to us in a rather general framework. See [30]. The goal of this appendix is to develop a minimal foundation of Olshanski’s spherical representation theory in the quantum setting in order to provide a basis for subsequent investigations of unitary representation theory for $U_q(\infty )$. It may be regarded as an answer to a question to the first version of this paper asked by Grigori Olshanski. In what follows, we will freely use standard facts in modular theory, see, e.g., [3, Sect. 2.5], [28, Chapter IV–IX]. Actually, the theory of standard forms based on modular theory lies behind the materials here. We also use the standard notation $(a\cdot \varphi \cdot b)(x)=\varphi (bxa)$ with a state $\varphi$ on a $C^*$-algebra A and $a,b,x\in A$.

Let G be a topological group. A triple $(T,{\mathcal {H}},\xi )$ is called a spherical representation if $(T,{\mathcal {H}})$ is a unitary representation of $G\times G$ and $\xi \in {\mathcal {H}}$ is a cyclic G-invariant unit vector, where we remark that G can be identified a subgroup of $G\times G$ by $g\in G\mapsto (g,g)\in G\times G$. Then, the function $\varphi :G\times G\rightarrow {\mathbb {C}}$ given as $\varphi (g,h):=\langle T(g,h)\xi ,\xi \rangle$ is called a spherical function. Namely, the spherical function $\varphi$ is a positive-definite continuous function satisfying that $\varphi (e)=1$ and $\varphi$ is G-biinvariant, that is, $\varphi (kgk',khk')=\varphi (g,h)$ for any $g,h,k,k'\in G$. It is known that there exists an affine bijective correspondence between the characters of G and the spherical functions. In this way, there also exists a correspondence between finite factor representations of G and irreducible spherical representations. See [17] for more details.

Let $G=(M,{\mathfrak {A}},\delta , R,\{\tau _t\}_{t\in {\mathbb {R}}})$ be a quantum group $W^*$-algebra (see Definition 3.1). We remark that the presence of dense $C^*$-algebra ${\mathfrak {A}}$ and comultiplication $\delta$ is not necessary in what follows. We denote by $M\otimes _\mathrm {bin} M$ the binormal tensor product of M, see [2, Sect. 4.3]. For any map f on $M\otimes _\mathrm {bin} M$, we define $f_L$ and $f_R$ on M by $f_L(x):=f(x\otimes 1)$, $f_R(x):=f(1\otimes x)$. If $f_L$ and $f_R$ are normal, then f is said to be binormal. Note that the binormal tensor product $M\otimes _\mathrm {bin}M$ enjoys the universality with respect to two commuting normal $*$-representations of M. We denote by $M_{\tau }$ the $\sigma$-weakly dense $*$-subalgebra of $\tau$-analytic elements.

Definition A.1

A triple $(T,{\mathcal {H}},\xi )$ is called a spherical representation if $(T,{\mathcal {H}})$ is a binormal $*$-representation of $M\otimes _\mathrm {bin} M$ and $\xi \in {\mathcal {H}}$ is a cyclic unit vector satisfying that

$$\begin{aligned} T(1\otimes x)\xi =T(\kappa (x)\otimes 1)\xi \end{aligned}$$

(A.1)

for any $x\in M_{\tau }$, where $\kappa :=R\circ \tau _{-\mathrm {i}/2}=\tau _{-\mathrm {i}/2}\circ R$. Then, $\xi$ is called a spherical vector. Two spherical representations $(T_i,{\mathcal {H}}_i,\xi _i)$ ($i=1,2$) are unitarily equivalent if there exists a unitary intertwiner U from $(T_1,{\mathcal {H}}_1)$ to $(T_2,{\mathcal {H}}_2)$ satisfying that $U\xi _1=\xi _2$.

Recall that $\kappa$ becomes the canonical antipode when G comes from a usual quantum group like $SU_q(2)$, etc., see [14, Remark 1.3(a)]. In this viewpoint, Eq. (A.1) can be regarded as an analog of $T(e,g)\xi =T(g^{-1},e)\xi$ for every $g\in G$ with a unitary representation $(T,{\mathcal {H}})$ of an ordinary group $G\times G$ and $\xi \in {\mathcal {H}}$. This is clearly equivalent to that $T(g,g)\xi =\xi$ for every $g\in G$. Therefore, our definition is a natural generalization of Olshanski’s one for ordinary groups.

Let $\chi$ be a quantized character of G and $(\pi _\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$ the associated GNS-triple. Since $\chi$ is a $\tau$-KMS state, $\xi _\chi$ is separating and cyclic for $\pi _\chi (M)$. This guarantees that $\chi$ “extends” to a faithful normal state ${\hat{\chi }}$ on $\pi _\chi (M)$, i.e., ${\hat{\chi }}(\pi _\chi (x))=\chi (x)$ for any $x\in M$. See [4, Corollary 5.3.8] and around there. Moreover, the modular automorphism group $\{\sigma ^{{\hat{\chi }}}_t\}_{t\in {\mathbb {R}}}$ associated with ${\hat{\chi }}$ satisfies that $\sigma ^{{\hat{\chi }}}_t(\pi _\chi (x))=\pi _\chi (\tau _t(x))$ for every $x\in M$ and $t\in {\mathbb {R}}$. Hence, the modular conjugation $J_{{\hat{\chi }}}:{\mathcal {H}}_\chi \rightarrow {\mathcal {H}}_{{\hat{\chi }}}$ is known to be given by the formula: $J_{{\hat{\chi }}}\pi _\chi (x)\xi _\chi =\pi _\chi (\tau _{-\mathrm {i}/2}(x^*))\xi _\chi$ for every $x\in M_\tau$. Then, it is well known that $J_{{\hat{\chi }}}\pi _\chi (M)J_{{\hat{\chi }}}$ coincides with the commutant $\pi _\chi (M)'$. By universality of binormal tensor product, there is a unique binormal $*$-representation $(T_\chi ,{\mathcal {H}}_\chi )$ of $M\otimes _\mathrm {bin} M$ such that $T_\chi (x\otimes y)=\pi _\chi (x)J_{{\hat{\chi }}}(\pi _\chi (R(y)^*))J_{{\hat{\chi }}}$ for any simple tensor $x\otimes y$.

Lemma A.1

The following three assertions hold true:

• (1) $(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$ is a spherical representation.

• (2) $(T_\chi ,{\mathcal {H}}_\chi )$ is irreducible if and only if $\chi$ is extreme; that is, $(\pi _\chi ,{\mathcal {H}}_\chi )$ is factorial.

• (3) $(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$ is a unique spherical representation up to unitarily equivalence satisfying that $\langle (T_\chi )_L(x)\xi _\chi ,\xi _\chi \rangle =\chi (x)$ for any $x\in M$.

Proof

The first and the second assertions are easy to prove. We leave them to the reader and prove only the third assertion. Let $(T,{\mathcal {H}},\xi )$ be a spherical representation satisfying $\langle T_L(x)\xi ,\xi \rangle =\chi (x)$ for any $x\in M$. By the uniqueness of GNS-triple associated with $\chi$, it suffices to show that $\xi$ is cyclic for $T_L(M)$. By Eq. (A.1), we have $T(x\otimes y)\xi =T_L(x\kappa (y))\xi$ for any $x\in M$ and $y\in M_\tau$. Thus, $T(M\otimes _\mathrm {bin} M)\xi \subset \overline{T_L(M)\xi }^{\Vert \,\cdot \,\Vert }$ since $M_\tau$ is $\sigma$-weakly dense in M and T is binormal. Namely, $\xi$ is also cyclic for $T_L(M)$. $\square$

For a spherical representation $(T,{\mathcal {H}},\xi )$, we define a state $\varphi :=\varphi _{(T,{\mathcal {H}},\xi )}$ on $M\otimes _\mathrm {bin} M$ by $\varphi (X):=\langle T(X)\xi ,\xi \rangle$ for any $X\in M\otimes _\mathrm {bin} M$, which we call the spherical function associated with $(T,{\mathcal {H}},\xi )$. Recall that original spherical functions are defined as certain positive-definite continuous functions on $G\times G$ with an ordinary group G in Olshanski’s theory. Our naive idea here is to translate positive-definite continuous functions on groups into positive linear functionals on corresponding $C^*$-algebra.

It is fairly elementary to check that the above $\varphi$ is binormal and satisfies that

$$\begin{aligned} (\kappa (x)\otimes 1)\cdot \varphi =(1\otimes x)\cdot \varphi \end{aligned}$$

(A.2)

for every $x\in M_\tau$. Here is an abstract definition of spherical functions:

Definition A.2

A binormal state $\varphi$ on $M\otimes _\mathrm {bin} M$ is called a spherical function if $\varphi$ satisfies Eq. (A.2).

The following lemma gives naturality of quantized characters from the viewpoint of spherical representation theory.

Lemma A.2

For any spherical function $\varphi$, the state $\varphi _L$ on M becomes a quantized character of G. Moreover, $(T_L,{\mathcal {H}},\xi )$ gives the GNS-triple associated with $\varphi _L$ if $\varphi$ is associated with a spherical representation $(T,{\mathcal {H}},\xi )$.

Proof

Since $\varphi _L$ is normal, it suffices to show that $\varphi _L$ is a $\tau$-KMS state. Since $\tau _{-i}=\kappa ^2$ and $\tau _{\mathrm {i}/2}(y)^*=\tau _{-\mathrm {i}/2}(y^*)$, we have

$$\begin{aligned} \varphi _L(x\tau _{-\mathrm {i}}(y))&=[(\kappa ^2(y)\otimes 1)\cdot \varphi ](x\otimes 1)\\&=[(1\otimes \kappa (y))\cdot \varphi ](x\otimes 1)\\&=\overline{[(1\otimes \kappa ^{-1}(y^*))\cdot \varphi ](x^*\otimes 1)}\\&=\overline{[(y^*\otimes 1)\cdot \varphi ](x^*\otimes 1)}\\&=\varphi _L(yx) \end{aligned}$$

for any $x\in M$ and $y\in M_{\tau }$. If $\varphi$ is associated with a spherical representation $(T,{\mathcal {H}},\xi )$, then it is clear that $\varphi _L(x)=\langle T_L(x)\xi ,\xi \rangle$ for any $x\in M$. Moreover, $\xi$ is cyclic for $T_L(M)$ (see the proof of Lemma A.1). Thus, $(T_L,{\mathcal {H}},\xi )$ is the GNS-triple associated with $\varphi$. $\square$

Remark A.1

In a similar way to Lemma A.2, we can prove that $\varphi _R(x\kappa ^{-2}(y))=\varphi _R(yx)$ for any $x\in M$ and $y\in M_\tau$. Since $\kappa ^{-2}=\tau _{i}$, we conclude that $\varphi _R$ is $\tau$-KMS state on M with the inverse temperature 1.

We are in a position to give a precise relationship between spherical functions and quantized characters. The next theorem can be understood as a quantum analog of one of the key facts in Olshanski’s spherical representation theory (see [13, Sect. 24]).

Theorem A.1

The following four assertions hold true:

• (1) The correspondence $(T,{\mathcal {H}},\xi )\mapsto \varphi _{(T,{\mathcal {H}},\xi )}$ gives a bijection from the unitarily equivalent classes of spherical representations to the spherical functions.

• (2) The correspondence $\chi \mapsto (T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$ gives a bijection from the spherical functions to the unitarily equivalent classes of spherical representations.

• (3) By (1), (2), the correspondence $\chi \mapsto \varphi _{(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )}$ gives a bijection from the spherical functions to the quantized characters. Moreover, the inverse correspondence is given as $\varphi \mapsto \varphi _L$.

• (4) Under the bijection in (2) (resp. in (1)), extreme quantized characters (resp. extreme spherical functions) correspond to irreducible spherical representations.

Proof

To prove (1), (2) and (3), it suffices to show the following three claims:

• (a) Any spherical representation $(T,{\mathcal {H}},\xi )$ is unitarily equivalent to $(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$, where $\chi :=\varphi _L$ and $\varphi :=\varphi _{(T,{\mathcal {H}},\xi )}$.

• (b) Any spherical function $\varphi$ is equal to $\varphi _{(T,{\mathcal {H}},\xi )}$, where $(T,{\mathcal {H}},\xi ):=(T_\chi ,{\mathcal {H}}_{\chi },\xi _{\chi })$ and $\chi :=\varphi _L$.

• (c) Any quantized character $\chi$ of G is equal to $\varphi _L$, where $\varphi :=\varphi _{(T,{\mathcal {H}},\xi )}$ and $(T,{\mathcal {H}},\xi ):=(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$.

Claims (a) and (c) clearly follow from Lemma A.1(1), (3) and Lemma A.2. We prove Claim (b). Let $\varphi$ be a spherical function. By Lemma A.2 and Lemma A.1(1), we obtain a quantized character $\chi :=\varphi _L$ of G and the spherical representation $(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )$. Then, we have $\langle T_\chi (x\otimes y)\xi _\chi ,\xi _\chi \rangle =\chi (x\kappa (y))=\varphi (x\kappa (y)\otimes 1)=\varphi (x\otimes y)$ for any $x\in M$ and $y\in M_\tau$. Since $M_\tau$ is $\sigma$-weakly dense in M and $T_\chi$ and $\varphi$ are binormal, we obtain $\varphi _{(T_\chi ,{\mathcal {H}}_\chi ,\xi _\chi )}=\varphi$.

The fourth assertion follows from Lemma A.1(2). $\square$

Next, we will give two propositions, which can be regarded as quantum analogs of other key facts in Olshanski’s spherical representation theory. For a $*$-representation $(T,{\mathcal {H}})$ of $M\otimes _\mathrm {bin} M$, we define

$$\begin{aligned}{\mathcal {H}}^0:=\{\eta \in {\mathcal {H}}\mid T(1\otimes x)\eta =T(\kappa (x)\otimes 1)\eta \text { for any }x\in M_{\tau }\}.\end{aligned}$$

The first one is rather easy to prove. Hence, we leave it to the reader.

Proposition A.1

Let $(T,{\mathcal {H}},\xi )$ be a spherical representation. Then, $(T,{\mathcal {H}})$ is irreducible if $\dim {\mathcal {H}}^0=1$.

The usual proof of the following fact in the original setting uses the notion of Gelfand pairs crucially, but our proof below uses the Connes Radon–Nikodym theorem instead. It seems an interesting question to formulate a quantum analog of Gelfand pairs (in this context).

Proposition A.2

Let $(T,{\mathcal {H}})$ be a $*$-representation of $M\otimes _\mathrm {bin} M$ such that $T_L$ and $T_R$ are normal. Then, $\dim {\mathcal {H}}^0\le 1$ if $(T,{\mathcal {H}})$ is irreducible.

Proof

Assume that $(T,{\mathcal {H}})$ is irreducible and ${\mathcal {H}}^0\ne \{0\}$. Let $\xi _i \in {\mathcal {H}}^0 (i=1,2)$ be unit vectors. It suffices to show that $\xi _2$ is proportional to $\xi _1$. Since $(T,{\mathcal {H}})$ is irreducible, $\xi _i$ is cyclic for $T(M\otimes _\mathrm {bin} M)$. Thus, $(T,{\mathcal {H}},\xi _i)$ is a spherical representation, and hence, by Lemma A.2, we obtain a quantized character $\chi _i$ of G and its GNS-triple $(T_L,{\mathcal {H}},\xi _i)$. By Lemma A.1(2), (3), the representation $(T_L,{\mathcal {H}})$ is factorial.

Recall that the state ${\hat{\chi }}_i$ on $T_L(M)$ given by ${\hat{\chi }}_i(T_L(x))=\chi _i(x)$ for any $x\in M$ is faithful normal and its modular automorphism group $\{\sigma ^{{\hat{\chi }}_i}_t\}_{t\in \mathbb {R}}$ is given by $\sigma ^{{\hat{\chi }}_i}_t(T_L(x))=T_L(\tau _t(x))$ for any $x\in M$ and $t\in \mathbb {R}$. Namely, $\sigma ^{{\hat{\chi }}_1}_t=\sigma ^{{\hat{\chi }}_2}_t$ for any $t\in \mathbb {R}$. Then, by [28, Theorem VIII.3.3(d)], the Connes Radon–Nikodym cocycle $\{(D{\hat{\chi }}_1:D{\hat{\chi }}_2)_t\}_{t\in \mathbb {R}}$ of ${\hat{\chi }}_1$ with respect to ${\hat{\chi }}_2$ must fall into the center of $T_L(M)$. Since $(T_L,\mathcal {H})$ is a factor representation, the center of $T_L(M)$ must be trivial, and thus, $\{(D{\hat{\chi }}_1:D{\hat{\chi }}_2)_t\}_{t\in \mathbb {R}}$ is a one-parameter group of scalar unitary operators, i.e., $(D\omega _1:D\omega _2)_t=\lambda ^{\mathrm {i}t}$ for some $\lambda >0$. Using the well-known uniqueness result for Connes Radon–Nikodym cocycle, we can prove that ${\hat{\chi }}_1=\lambda {\hat{\chi }}_2$. Evaluating this equation at 1, we obtain that $\lambda =1$, that is, ${\hat{\chi }}_1={\hat{\chi }}_2$ and hence $\chi _1=\chi _2$. By Theorem A.1, $(T,\mathcal {H},\xi _1)$ and $(T,\mathcal {H},\xi _2)$ are unitarily equivalent. Since $(T,\mathcal {H})$ is irreducible, any unitary intertwiner from $(T,\mathcal {H},\xi _1)$ to $(T,\mathcal {H},\xi _2)$ must be a scalar operator; that is, $\xi _2$ is proportional to $\xi _1$. $\square$