Quadratic Subproduct Systems, Free Products, and Their C*-Algebras

Abstract

Motivated by the interplay between quadratic algebras, noncommutative geometry, and operator theory, we introduce the notion of quadratic subproduct systems of Hilbert spaces. Specifically, we study the subproduct systems induced by a finite number of complex quadratic polynomials in noncommuting variables, and describe their Toeplitz and Cuntz–Pimsner algebras. Inspired by the theory of graded associative algebras, we define a free product operation in the category of subproduct systems and show that this corresponds to the reduced free product of the Toeplitz algebras. Finally, we obtain results about the K-theory of the Toeplitz and Cuntz–Pimsner algebras of a large class of quadratic subproduct systems.

Introduction

The study of subproduct systems and their C*-algebras has become a significant area of research at the intersection of multivariate operator theory [6], noncommutative geometry, and operator algebras. First introduced by Shalit and Solel in [29], and around the same time by Bhat and Mukherjee in the Hilbert space setting [8], subproduct systems provide a natural framework for understanding row-contractive tuples of operators subject to polynomial constraints.

In this paper, we focus on quadratic subproduct systems, which are subproduct systems of Hilbert spaces arising from a finite set of quadratic polynomials in a finite number of noncommuting variables. This class exhibits rich algebraic and operator-theoretic properties, and is quite a natural one to consider, given that algebras are often given in terms of commutation rules between their generators. Indeed, noncommutative algebras defined by quadratic relations are crucial examples of noncommutative spaces, such as those appearing in Manin’s programme for noncommutative geometry [21, 22]. Such quadratic algebras include the deformations of quantum groups—and spaces—arising from an R-matrix, as defined in the seminal work of Faddeev, Reshetikhin, and Takhtajan [10]. These continue to play a central role in noncommutative geometry, providing a rich source of examples of noncommutative spaces.

The interaction between subproduct systems and both classical and quantum groups extends beyond the construction of the former, offering insights into the algebraic, geometric, and topological aspects of the underlying noncommutative spaces [4, 5, 14, 15]. The presence of quantum group symmetries allows for elegant computations of the K-theoretic invariants of their C*-algebras. More recently, Aiello, Del Vecchio, and Rossi have introduced a subproduct system of finite-dimensional Hilbert spaces associated to the Motzkin planar algebra [1], generalising the Temperley–Lieb subproduct systems of Habbestad and Neshveyev [14, 15].

Building on the framework established in previous studies, we examine the subproduct system analogue of the free product construction for noncommutative associative algebras. One of our motivations, in addition to the naturality of the free product construction, comes from the representation theory of the quantum group $S{U}_q(2)$. Our starting point is the observation that a free-product structure naturally appears when applying the algorithm in [5, Section 2] to multiplicity-free representations. This feature allows us to derive new insights into the algebraic and analytical properties of such subproduct systems, particularly in the context of their Fock spaces and associated C$_{}^{\ast }$-algebras.

The structure of the paper is as follows: We start by recalling the basic definitions and constructions for subproduct systems in Section 2. In Section 3, we introduce quadratic subproduct systems, highlighting their connections with quadratic algebras. We also define quadratic subproduct systems with few relations and discuss how they relate to generic quadratic algebras, for which a lot is known about their growth and Hilbert series. Section 4 is devoted to the free product operation on quadratic subproduct systems. Here, we establish explicit formulas for the fibres of the free product and describe their fusion rules. Our main result for the section, Proposition 4.8, is a decomposition theorem for the Fock space of the free product of subproduct systems.

In Section 5, we study the Toeplitz algebras associated with these free products. Theorem 5.4 asserts that the free product structure is preserved at the level of Toeplitz algebras, more precisely in terms of a reduced free product. Moreover, Theorem 5.13, stated below, ensures that the free product construction allows us to bootstrap properties such as nuclearity and KK-equivalence to the complex numbers from smaller building blocks.

Theorem

Let $\mathcal{H}$ and $\mathcal{K}$ be quadratic subproduct systems of Hilbert spaces. Assume that the Toeplitz algebras ${\mathbb{T}}_{\mathcal{H}}$ and ${\mathbb{T}}_{\mathcal{K}}$ are both nuclear and KK-equivalent to the complex numbers. Then so is the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}$.

We also demonstrate how our free product construction applied to monomial quadratic ideals corresponds to the graph join operation at the level of Cuntz–Pimsner algebras.

Section 6 focusses on Temperley–Lieb subproduct systems, a subclass of quadratic subproduct systems defined by specific combinatorial constraints, introduced in [14] and further studied in [4, 15]. We analyse the free products of Temperley–Lieb subproduct systems, compute their K-theory, and construct explicit KK-equivalences for their Toeplitz algebras. We conclude the paper by studying the subproduct system of a finite-dimensional multiplicity-free unitary $S{U}_q(2)$-representation, answering some questions regarding their structure and K-theory that were left open in [5].

Preliminaries on Subproduct Systems and Their Algebras

We start this section by recalling some basic facts from the theory of subproduct systems of Hilbert spaces and their C$_{}^{\ast }$-algebras. Our main references are [29, 33]. Although, in their original paper, Shalit and Solel studied subproduct systems in the more general setting of C$_{}^{\ast }$-and W$_{}^{\ast }$-correspondences, we shall focus here on the Hilbert space case.

By a subproduct system of finite-dimensional Hilbert spaces, we shall mean a sequence of finite-dimensional Hilbert spaces $\mathcal{H}={\{H_n\}}_{n\in {\mathbb{N}}_0}$, with $dim(H_0)=1$, together with isometries

$${\iota }_{m,n}:H_{m+n}\to H_m\otimes H_n,$$

satisfying

$$\begin{array}{c}\hfill ({\iota }_{m,n}\otimes 1){\iota }_{m+n,k}=(1\otimes {\iota }_{n,k}){\iota }_{m,n+k}:H_{m+n+k}\to H_m\otimes H_n\otimes H_k,\end{array}$$

for all $m,n,k\in {\mathbb{N}}_0$, where 1 denotes the identity operator.

A subproduct system is called standard if $H_0=\mathbb{C}$, $H_{m+n}\subseteq H_m\otimes H_n$, and the maps ${\iota }_{m,n}$ agree with the embedding maps.

As pointed out in [29], standard subproduct systems of finite-dimensional Hilbert spaces provide the natural framework for studying row-contractive tuples of operators subject to polynomial constraints, as made transparent by the existence of a noncommutative Nullstellensatz.

Proposition 2.1

([29, Proposition 7.2]) Let H be a d-dimensional Hilbert space. Then there is a bijective inclusion-reversing correspondence between the proper homogeneous ideals $J\subset \mathbb{C}⟨X_1,\dots ,X_d⟩$ and the standard subproduct systems ${\{H_n\}}_{n\in {\mathbb{N}}_0}$ with $H_1\subseteq H$.

Let us fix an orthonormal basis ${\{e_i\}}_{i=1}^d$ for H. For a noncommutative polynomial $P=\sum c_{\alpha }{X}^{\alpha }$ in variables $X_1\cdots ,X_d$, we write $P(e)=\sum c_{\alpha }{e}_{\alpha }$, where $e_{\alpha }=e_{{\alpha }_1}\otimes \cdots \otimes e_{{\alpha }_k}$ for $\alpha ={\alpha }_1\cdots {\alpha }_k$ a length-k word. The correspondence works as follows:

To any proper homogeneous ideal $J\subset \mathbb{C}⟨X_1,\dots ,X_d⟩$, one associates the standard subproduct system $H^J$ with fibres $H_n:=H^{\otimes n}\ominus \{P(e)|P\in J^{(n)}\}$, for every $n\ge 0$, where $J^{(n)}$ denotes the degree-n component of the ideal J.

Following [29, Definition 7.3], we refer to ${\mathcal{H}}^J$ and $J_{\mathcal{H}}$ as the subproduct system associated with the ideal J and the ideal associated with the subproduct system $\mathcal{H}$, respectively.

While, in principle, the above construction depends on the choice of an orthonormal basis for H, different choices yield isomorphic subproduct systems in the sense of [29, Definition 1.4].

Proposition 2.2

([29, Proposition 7.4]) Let $\mathcal{H}$ and $\mathcal{K}$ be standard subproduct systems with $dim(H_1)=dim(K_1)=d<\infty$. Then $\mathcal{H}$ is isomorphic to $\mathcal{K}$ if and only if there is a unitary linear change of variables in $\mathbb{C}⟨X_1,\dots ,X_d⟩$ that sends $J_{\mathcal{H}}$ onto $J_{\mathcal{K}}$.

In a basis-independent fashion, Proposition 2.1 can also be formulated as follows: There is a bijective inclusion-reversing correspondence between the proper homogeneous ideals J of the free algebra in $dim(H)$-generators and the standard subproduct systems ${\{H_n\}}_{n\in {\mathbb{N}}_0}$ with $H_1\subseteq H$.

It is worth recalling that all standard subproduct systems of finite-dimensional Hilbert spaces are obtained this way, see [29, Proposition 7.2]. As we mentioned in the introduction, we will focus on standard subproduct systems induced by a finite number of quadratic polynomials in noncommuting variables, as these form a more tractable class of examples.

Toeplitz and Cuntz–Pimsner Algebras of Subproduct Systems

We conclude this section by recalling the construction of the Toeplitz and Cuntz–Pimsner algebras of a subproduct system of Hilbert spaces.

The Fock space of the subproduct system $\mathcal{H}$ is the direct sum Hilbert space

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathcal{H}}:=\underset{n\ge 0}{⨁}{H}_n.\end{array}$$

On the Hilbert space ${\mathcal{F}}_{\mathcal{H}}$ we consider operators defined by

$$\begin{array}{c}\hfill T_{\xi }(\zeta ):={\iota }_{1,n}^{\ast }(\xi \otimes \zeta ),\xi \in H_1,\zeta \in H_n.\end{array}$$

Note that the Fock space is a subspace of the full Fock space of $H_1$:

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathcal{H}}\subseteq \underset{n\ge 0}{⨁}{H}_1^{\otimes n}\text{and}{T}_{\xi }(\zeta )=f_{n+1}(\xi \otimes \zeta ),\xi \in H_1,\zeta \in H_n,\end{array}$$

where $f_{n+1}$ is the projection $f_{n+1}:H_1^{\otimes (n+1)}\to H_{n+1}$.

The Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}}$ of the subproduct system $\mathcal{H}$ is the unital C$_{}^{\ast }$-algebra generated by $T_1,T_2,...,T_d$, where $T_i=T_{{\xi }_i}$ for an orthonormal basis ${({\xi }_i)}_{i=1}^d$ of $H_1$. If one denotes by $e_0$ the rank-one projection onto $H_0$, it is straightforward to verify that

$$\begin{array}{c}\hfill 1_{\mathcal{F}}-\sum _{i=1}^d{T}_i{T}_i^{\ast }=e_0.\end{array}$$

Consequently, the compact operators on the Fock space $\mathbb{K}(\mathcal{F}(\mathcal{H}))$ are contained in ${\mathbb{T}}_{\mathcal{H}}$ (cf. [33, Corollary 3.2]). This fact is used to define the Cuntz–Pimsner algebra ${\mathbb{O}}_{\mathcal{H}}$ of the subproduct system as the quotient:

1

Quadratic Subproduct Systems from Quadratic Algebras

Quadratic Algebras and Their Hilbert Series

In this work, we shall use several results from the theory of quadratic algebras, particularly in connection with their Hilbert series. Our main references are [27, 31]. Our base field will be the complex numbers.

Given a vector space V, we denote its tensor algebra by $\mathcal{T}(V)$. This is naturally graded by rank, and we write

$$\begin{array}{c}\hfill \mathcal{T}(V):=\underset{n\ge 0}{⨁}{\mathcal{T}}^n(V),{\mathcal{T}}^n(V):=V^{\otimes n}.\end{array}$$

Definition 3.1

A graded algebra $A\simeq {⨁}_{n\ge 0}{A}_n$ is called one-generated if the natural map $p:\mathcal{T}(A_1)\to A$ from the tensor algebra generated by $A_1$ is surjective. We call a one-generated algebra quadratic if the ideal $J_A:=ker(p)$ is generated, as a two-sided ideal, by

$$\begin{array}{c}\hfill I_A:=J_A\cap {\mathcal{T}}^2(A_1)\subseteq A_1\otimes A_1.\end{array}$$

In other words, a quadratic algebra is determined by a vector space of generators $V=A_1$ and a subset of relations $I_A\subset A_1\otimes A_1.$ We shall denote with R the complex vector subspace of $A_1\otimes A_1$ spanned by the relations. If $dim(R)=r$, we call A an r-relator quadratic algebra.

Recall that for a graded vector space $V={⨁}_{k\ge 0}{V}_k$, with finite-dimensional graded components, its Hilbert series is the formal power series

$$\begin{array}{c}\hfill h_V(z)=\sum _{k\ge 0}dim(V_k)z^k.\end{array}$$ 2

Hilbert series of associative algebras provide information about their growth. In [2], Anick studied them under certain finiteness hypotheses by considering a well-ordering defined as follows. Given two formal power series $f(z),g(z)\in \mathbb{R}[[z]]$, we write $f(z)\ge g(z)$ if the inequality holds coefficient-wise. Moreover, we write $|f(z)|$ for the series obtained by deleting all the terms starting from the first negative term. Using this notation, one can write a lower bound for the Hilbert series of a quadratic algebra:

Proposition 3.2

[3, Proposition 2.3] For any quadratic algebra in m generators and r relations, the Hilbert series satisfies

$$\begin{array}{c}\hfill h_A(z)\ge |{(1-mz+r{z}^2)}^{-1}|.\end{array}$$

Theorem 3.3

(cf. [27, Proposition 4.1]) Let A be a graded quadratic algebra with $dim(A_1)=d$ and $dim(A_2)=s$. The minimal possible value for $dim(A_3)$ is

$$\left\{\begin{array}{cc}0\hfill & \text{if}s\le \frac{d^2}{2},\hfill \\ 2ds-d^3\hfill & \text{if}s\ge \frac{d^2}{2}.\hfill \end{array}\right)$$

In his work, Anick also answered the question of which algebras attain the minimal Hilbert series by considering the notion of genericity. Let us first clarify what we mean by the term generic.

Definition 3.4

(cf. [27, Chapter 6]) A complex generic quadratic algebra in d generators and r relations is a generic point in the variety ${\mathcal{Q}}_{d,r}$ of quadratic algebras with $dim(A_1)=d$ and $dim(R)=r$.

By definition, such an r-relator quadratic algebra is determined by a $(d^2-r)$-dimensional subspace of ${\mathbb{C}}^2$. We can therefore identify the variety ${\mathcal{Q}}_{d,r}$ of such quadratic algebras with the complex Grassmannian ${\mathbf{Gr}}_{\mathbb{C}}(r,d^2)$. One then says that a generic complex quadratic algebra is a quadratic algebra corresponding to a generic point in the Grassmannian variety of quadratic algebras (cf. [2, Lemma 4.1]).

Anick’s main result establishes that generic algebras are exactly those which possess the coefficient-wise minimal Hilbert series [2, Definition 4.9]. Additionally, in the quadratic case, adding some further constraints on the number of generators and relations allows one to obtain an explicit formula for the generic Hilbert series:

Proposition 3.5

([27, Proposition 4.2]) A generic quadratic algebra A in d generators and r relations is Koszul1 if and only if one of the following inequalities holds:

$$\begin{array}{c}\hfill r\ge \frac{3d^2}{4},r\le \frac{d^2}{4}.\end{array}$$

Then the Hilbert series of A is either

$$\begin{array}{ccc}\hfill h_A(z)& =1+dz+(d^2-r)z^2,\hfill & \hfill \text{or}\end{array}$$ 3

$$\begin{array}{cc}\hfill h_A(z)& ={(1-dz+r{z}^2)}^{-1},\hfill \end{array}$$ 4

respectively.

Note that, in the first case, when $r\ge 3d^2/4$, (3) implies that the quadratic algebra is finite-dimensional (and in particular $dim(A_3)=0$). This should serve as motivation for focusing on the constraint $r\le d^2/4$ later on. In that case, we shall talk of a quadratic algebra with few relations. Let us stress that, by Anick’s Theorem, a quadratic algebra A with few relations is generic if and only if its Hilbert series equals (4).

Remark 3.6

In the literature, one can encounter another related notion of having few relations, due to Zhang [35]. The main result states that whenever $\text{rank}(R)>dim(R)+1$, the quadratic algebra is Koszul with global dimension 2, and its Hilbert series is given by (4), making such algebras automatically generic.

Hilbert series of general quadratic algebras with a fixed number of generators and relations are well-studied and understood, at least in low dimensions. We refer the reader to [27, Section 6.5] for some explicit expressions of the Hilbert series.

Example 3.7

When $d=2$ and $r=1$, a non-generic quadratic algebra must necessarily have the following Hilbert series:

$$\begin{array}{c}\hfill {(1-z-z^2)}^{-1}=1+2z+3z^2+5z^3+8z^4+13z^5+\cdots .\end{array}$$ 5

An example is the algebra $\mathbb{C}⟨X_1,X_2⟩/⟨X_1^2⟩$, the quotient of the free algebra in two variables by the quadratic monomial ideal generated by $X_1^2$. We will re-encounter this algebra in Example 3.19.

Free Products of Quadratic Algebras and Their Hilbert Series

The category of unital graded algebras over a field has a natural coproduct operation, given by the algebraic free product.

Definition 3.8

Given two graded algebras A and B over a field k, their free product, denoted $A\bigsqcup B$, is defined as the associative algebra generated freely by A and B. Explicitly:

$$\begin{array}{c}\hfill A\bigsqcup B:=\underset{i\ge 0,{ϵ}_1,{ϵ}_2\in \{0,1\}}{⨁}{A}_{+}^{{ϵ}_1}\otimes {(B_{+}\otimes A_{+})}^{\otimes i}\otimes B_{+}^{{ϵ}_2},\end{array}$$ 6

with the usual convention that $A_{+}:={⨁}_{n\ge 1}{A}_n$.

It is natural to wonder how the Hilbert series behaves when one considers free products. Given two finitely-presented algebras A and B, by [31, Theorem 4.5.3], the Hilbert series of the free product algebra $A\bigsqcup B$ can be expressed in terms of the Hilbert series of $h_A(z)$ and $h_B(z)$ of the algebras A and B:

$$\begin{array}{c}\hfill h_{A\bigsqcup B}{(z)}^{-1}=h_A{(z)}^{-1}+h_B{(z)}^{-1}+1.\end{array}$$ 7

This has important consequences for the question of genericity.

Remark 3.9

Suppose that A and B are $r_0$- and $r_1$-relator generic quadratic algebras in $d_0$ and $d_1$ generators, respectively, satisfying the additional condition $r_i\le d_i^2/4,$ for $i=0,1$. The formula for the Hilbert series of the free product of algebras (7) yields

$$\begin{array}{c}\hfill h_{A\bigsqcup B}(z)={(1-(d_0+d_1)z+(r_0+r_1)z^2)}^{-1},\end{array}$$

implying that $A\bigsqcup B$ is a generic quadratic algebra in $d_0+d_1$ generators and $r_0+r_1$ relations.

Other important operations and constructions that preserve the class of quadratic algebras are Veronese powers and Segre products [27]. We defer the treatment of their operator algebraic counterparts to future work.

Quadratic Subproduct Systems

Having discussed the fundamentals of the theory of quadratic algebras, we are ready to introduce quadratic subproduct systems of Hilbert spaces.

Recall first the definition of the maximal suproduct system with prescribed fibres up to a finite fixed level.

Definition 3.10

([29, Section 6]) Let $H_0=\mathbb{C}$, and let $H_1$ be a Hilbert space. Let $H_i$,$i=2,\cdots ,k$ be subspaces of $H_1^{\otimes i}$ respectively. The maximal subproduct system with prescribed fibres up to level k is defined as the subproduct system $\mathcal{H}={\{H_n\}}_{n\ge 0}$ with fibers inductively defined as

$$\begin{array}{c}\hfill H_n=\bigcap _{i+j=n}{H}_i\otimes H_j,n>k.\end{array}$$

Proposition 3.11

Let $\mathcal{H}$ be a standard subproduct system of finite-dimensional Hilbert spaces with $dim(H_1)=d$. The following facts are equivalent:

1. There exists $R\subseteq H_1\otimes H_1$ such that H is isomorphic to the maximal subproduct system with fibres $\mathbb{C},H_1,R^{\perp }$ in the sense of Definition 3.10.

2. There exist finitely many homogeneous quadratic polynomials $f_1,\cdots ,f_k$ in $\mathbb{C}⟨X_1,\dots ,X_d⟩$ such that $\mathcal{H}$ is isomorphic to the subproduct system $H^{⟨f_1,\cdots ,f_k⟩}$ associated to the ideal $⟨f_1,\cdots ,f_k⟩$ generated by those polynomials.

3. The ideal $J_{\mathcal{H}}$ is generated by a finite number of quadratic polynomials.

A standard subproduct system of finite-dimensional Hilbert spaces satisfying any of the above conditions will be called a quadratic subproduct system of Hilbert spaces.

Definition 3.12

A quadratic subproduct system with $dim(H_1)=d$ and $dim(R)$ $=r$ will be called an r-relator quadratic subproduct system. If R corresponds to a generic point of the Grassmannian ${\mathbf{Gr}}_{\mathbb{C}}(r,d^2)$, then we call $\mathcal{H}$ a generic r-relator quadratic subproduct system.

Note that everything still makes sense in the case $r=0$, as one obtains product systems of finite-dimensional Hilbert spaces. Everything we discuss in the rest of this work holds in that setting as well.

A special class of examples is that of one-relator quadratic subproduct systems, i.e., those subproduct systems whose underlying ideal $J_{\mathcal{H}}$ is generated by a single quadratic polynomial. In that case, we have the following result due to Shalit and Solel.

Theorem 3.13

([29, Proposition 11.1]) Let $A,B\in M_d(\mathbb{C})$. Consider the two quadratic subproduct systems ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ given by the polynomials

$$\begin{array}{c}\hfill \sum _{i,j=1}^d{A}_{\mathit{ij}}{X}_i{X}_j,\sum _{i,j=1}^d{B}_{\mathit{ij}}{X}_i{X}_j,\end{array}$$

respectively. Then there is an isomorphism $V:{\mathcal{H}}_A\to {\mathcal{H}}_B$ if and only if there exists $\lambda \in \mathbb{C}$ and a unitary $d\times d$ matrix U such that $B=\lambda U^{t}AU$.

In the following, when talking about the Hilbert series of a subproduct system of Hilbert spaces, we will mean the formal power series (2) of the underlying graded vector space.

Motivated by Proposition 3.5, we give the following definition.

Definition 3.14

Let $\mathcal{H}$ be an r-relator quadratic subproduct system of finite-dimensional Hilbert spaces with $dim(H_1)=d$. We say that $\mathcal{H}$ has few relations if

$$\begin{array}{c}\hfill r\le \frac{d^2}{4}.\end{array}$$ 8

Remark 3.15

For any quadratic subproduct system of finite-dimensional Hilbert spaces $\mathcal{H}$,

$$\begin{array}{c}\hfill H_{m+1}=H_1\otimes H_m\cap H_m\otimes H_1,\text{for all}m\ge 2.\end{array}$$ 9

Indeed, a quadratic subproduct system is a maximal standard subproduct system with prescribed fibres $H_1$ and $H_2$, for which the higher-level fibres are given as in Definition 3.10. Applying this to $H_m\otimes H_1$ and $H_1\otimes H_m$, we obtain

$$\begin{array}{c}\hfill H_m\otimes H_1=\bigcap _{k+l=m}{H}_k\otimes H_l\otimes H_1,\\ \hfill H_1\otimes H_m=\bigcap _{k+l=m}{H}_1\otimes H_k\otimes H_l,\end{array}$$

from which we obtain that

$$\begin{array}{c}\hfill H_i\otimes H_{m+1-j}=H_i\otimes H_{m-i}\otimes H_1\cap H_1\otimes H_{i-1}\otimes H_{m+1-i},\end{array}$$

since $H_i\subset H_1\otimes H_{i-1}$ and $H_{m+1-i}\subset H_{m-i}\otimes H_1$. By simple linear algebra arguments, (9) follows.

Remark 3.16

One may wonder whether the definition of a quadratic subproduct system could be extended to the setting of correspondences. This is clearly the case if we use Condition 1 in Proposition 3.11 and consider maximal subproduct systems with prescribed fibres in degrees up to two. However, two important aspects require additional care. Firstly, in the correspondence case, one loses the connection to the theory of polynomials in non-commuting variables. Most importantly, there are some technical issues related to the complementability of submodules, deeply connected to the theory of two-projections in Hilbert modules [24]. We postpone the treatment of quadratic subproduct systems of correspondences to future work.

The subproduct systems associated with an irreducible unitary SU(2)-representation from [5], and more generally the Temperley–Lieb subproduct systems studied in [4, 14, 15] are clear examples of one-relator quadratic subproduct systems, being defined by a single quadratic polynomial.

The Importance of Being Generic

As anticipated, we shall focus on generic subproduct systems with few relations, that is, satisfying $r\le d^2/4$.

Proposition 3.17

Let $\mathcal{H}$ be a generic quadratic subproduct system that satisfies the assumptions in Definition 3.14. Denote ${\delta }_n=dim(H_n)$. The sequence ${\{{\delta }_n\}}_{n\ge 0}$ satisfies the following recurrence relation:

$$\begin{array}{c}\hfill {\delta }_0=1;{\delta }_1=d;{\delta }_{n+1}=d·{\delta }_n-r·{\delta }_{n-1},n\ge 1.\end{array}$$ 10

Proof

The proof relies on a standard argument that involves the logarithmic derivative of the generating function for the Hilbert series of $\mathcal{H}$. $\square$

As a consequence, we have the following:

Corollary 3.18

Let $\mathcal{H}$ be a generic quadratic subproduct system that satisfies the assumptions in Definition 3.14. Then, for every $n\ge 1$, there are vector space isomorphisms

$$\begin{array}{c}\hfill H_n\otimes H_1\cong H_{n+1}\oplus H_{n-1}^{\oplus r},\end{array}$$

Finding explicit formulas for isometries that implement the above isomorphism may be a hard task. This is possible in some cases, including those we will encounter in Sections 6 and 7.

Example 3.19

(The Fibonacci subproduct system) Consider the subproduct system associated to the ideal $I:=⟨X_1^2⟩\subseteq \mathbb{C}⟨X_1,X_2⟩$. It is easy to see that it has Hilbert series (5), and as such, it cannot be generic. The dimension sequence of this subproduct system is the (shifted) Fibonacci sequence:

$$\begin{array}{c}\hfill {\delta }_0=1,{\delta }_1=2,{\delta }_{n+1}={\delta }_n+{\delta }_{n-1},n\ge 1.\end{array}$$

Free Products of Subproduct Systems and Their Fock Spaces

We are ready to define the free product of quadratic subproduct systems:

Definition 4.1

Let $\mathcal{H}={\{H_n\}}_{n\in {\mathbb{N}}_0}$ and $\mathcal{K}={\{K_n\}}_{n\in {\mathbb{N}}_0}$ be two quadratic subproduct systems. We then define the free product $\mathcal{H}\star \mathcal{K}$ of $\mathcal{H}$ and $\mathcal{K}$ as the maximal subproduct system with prescribed fibres

$$\begin{array}{c}\hfill {(H\star K)}_1:=H_1\oplus K_1,{(H\star K)}_2:={(H_2^{\perp }\oplus K_2^{\perp })}^{\perp }.\end{array}$$

In the polynomial picture, if we write $J_{\mathcal{H}}$ and $J_{\mathcal{K}}$ for the corresponding ideals, then the free product is obtained by considering the ideal generated by the disjoint union of the quadratic polynomials that generate $J_{\mathcal{K}}$ and $J_{\mathcal{K}}$ in the free algebra in $dim(H_1)+dim(K_1)$ variables.

We shall use Remark 3.15 to describe the Hilbert space ${(H\star K)}_2$ explicitly.

Lemma 4.2

Let $\mathcal{H}$ and $\mathcal{K}$ be two quadratic subproduct systems of finite-dimensional Hilbert spaces. We have an isomorphism of inner product spaces

$${(H\star K)}_2\simeq H_2\oplus K_2\oplus (H_1\otimes K_1)\oplus (K_1\otimes H_1).$$

We are interested in describing the fibres of the free product of quadratic subproduct systems using the formula from Remark 3.15, which involves tensor products and intersections. Let us first recall some enumerative combinatorics.

Definition 4.3

A composition of a positive integer $n\ge 1$ is a sequence ${\sigma }_1,\cdots ,{\sigma }_p$, ${\sigma }_i\in {\mathbb{Z}}_{+}$ with ${\sum }_{i=1}^p{\sigma }_i=n$. The ${\sigma }_i$’s are called the parts of n. We denote the set of compositions of n with $\mathcal{C}(n)$ and the subset of compositions of n into exactly p parts by ${\mathcal{C}}_p(n)$.

A given integer $n\ge 1$ has $2^{n-1}$ compositions. Moreover, for every $1\le p\le n$ the cardinality of ${\mathcal{C}}_p(n)$ equals the binomial coefficient $\left(\begin{array}{c}n-1\\ p-1\end{array}\right)$.

Let ${\mathcal{H}}^{(0)}$ and ${\mathcal{H}}^{(1)}$ be two subproduct systems of finite-dimensional Hilbert spaces, quadratic or not. For a fixed $m\in {\mathbb{N}}_0$, $1\le p\le m$ and $\underset{\_}{d}\in {\mathcal{C}}_p(m)$ and $j=0,1$, we define

$$H_{\underset{\_}{d}}^{(j)}:=H_1^{(j)}\otimes H_2^{((j+1)mod2)}\otimes \cdots \otimes H_p^{((j+p+1)mod2)}.$$

As already mentioned, we are interested in understanding how such spaces behave for tensor products and intersections, since those are the operations involved in the construction of a quadratic subproduct system, and in general, in the definition of a subproduct system with prescribed fibres.

Proposition 4.4

Let $\underset{\_}{d}\in C_p(m)$, $\underset{\_}{f}\in C_q(m)$. With the above notation, the intersection

$$\begin{array}{c}\hfill (H_1^{(i)}\otimes H_{\underset{\_}{d}}^{(j)})\cap (H_{\underset{\_}{f}}^{(l)}\otimes H_1^{(k)})=\mathrm{\varnothing },\end{array}$$

whenever $i\ne l$, $q\ne p,p-1,p+1$.

Moreover, the intersection $(H_1^{(i)}\otimes H_{\underset{\_}{d}}^{(j)})\cap (H_{\underset{\_}{f}}^{(i)}\otimes H_1^{(k)})$ is isomorphic to

$$\left\{\begin{array}{cc}{H}_{(d_1+1,d_2,\cdots ,1)}^{(i)}\hfill & i=j,q=p-1,k=p+i+1mod2,d_p=1,\hfill \\ \hfill & f_1=d_1+1,f_2=d_2,\cdots f_{p-1}=d_{p-1};\hfill \\ H_{(d_1+1,d_2,\cdots ,d_p)}^{(i)}\hfill & i=j,q=p,k=p+i+1mod2,d_p>1,\hfill \\ \hfill & f_1=d_1+1,f_2=d_2,\cdots f_p=d_p-1;\hfill \\ H_{(1,d_1,d_2,\cdots ,d_{p-1},1)}^{(i)}\hfill & i\ne j,q=p,d_p=1,\hfill \\ \hfill & f_1=1,f_2=d_1,\cdots ,f_p=d_{p-1};\hfill \\ H_{(1,d_1,d_2,\cdots ,d_{p-1},d_p)}^{(i)}\hfill & i\ne j,q=p+1,d_p>1\hfill \\ \hfill & f_1=1,f_2=d_1,\cdots ,f_p=d_{p-1},f_{p+1}=d_p+1;\hfill \\ \mathrm{\varnothing }\hfill & \text{otherwise}\hfill \end{array}\right)$$

Proof

Let us first suppose that $i=j$, then we are looking at the subspace

$$H_1^{(i)}\otimes H_{\underset{\_}{d}}^{(i)}\cap H_{\underset{\_}{f}}^{(i)}\otimes H_1^{(k)},$$

which we can rewrite as the intersection between

$$\begin{array}{c}\hfill H_1^{(i)}\otimes H_1^{(i)}\otimes H_2^{((i+1)mod2)}\otimes \cdots \otimes H_p^{((p+i+1)mod2)}\end{array}$$

and

$$\begin{array}{c}\hfill H_1^{(i)}\otimes H_2^{((i+1)mod2)}\otimes \cdots H_{f_q}^{((q+i+1)mod2)}\otimes H_1^{(k)}.\end{array}$$

First of all, we observe that if $p=1$, the intersection amounts to

$$H_1^{(i)}\otimes H_1^{(i)}\cap H_1^{(i)}\otimes H_1^{(k)}=\left\{\begin{array}{cc}{H}_{d+1}^{(i)}\hfill & f=d>1,k=j=i,\hfill \\ \mathrm{\varnothing }\hfill & \text{otherwise.}\hfill \end{array}\right)$$

If $p>1$, the non-triviality of the intersection forces $\underset{\_}{d}$ and $\underset{\_}{f}$ to satisfy either

$$d_1+1=f_1,d_2=f_2,\cdots d_{p-1}=f_q,d_p=1,$$

for $q=p-1$, $k=(p+i+1)mod2$; or

$$d_1+1=f_1,d_2=f_2,\cdots d_{p-1}=f_{q-1},d_p=f_q+1>1,$$

for $p=q$ and $k=(p+i+1)mod2$.

Similar considerations give the claim for the intersections for $i\ne j$. $\square$

With Proposition 4.4 in place, we can now provide an explicit description of the fibres of the free product of two quadratic subproduct systems.

Proposition 4.5

Let ${\mathcal{H}}^{(0)}$ and ${\mathcal{H}}^{(1)}$ be two quadratic subproduct systems. Their free product ${\mathcal{H}}^{(0)}\star {\mathcal{H}}^{(1)}$ satisfies

$$\begin{array}{c}\hfill {(H^{(0)}\star H^{(1)})}_m=\underset{i\in \{0,1\}}{⨁}\underset{p=1}{\overset{m}{⨁}}\underset{\underset{\_}{d}\in {\mathcal{C}}_p(m)}{⨁}{H}_{\underset{\_}{d}}^{(i)}.\end{array}$$ 11

Taking the free product of two quadratic subproduct systems is an associative operation:

Lemma 4.6

Let ${\{H_m^{(i)}\}}_{m\in {\mathbb{N}}_0}$, $i=0,1,2$ be three quadratic subproduct systems of finite-dimensional Hilbert spaces. Then for all $m\ge 0$, we have unitary isomorphisms

$$\begin{array}{c}\hfill ((H^{(1)}\star H^{(2)})\star H^{(3)}){)}_m={(H^{(1)}\star (H^{(2)}\star H^{(3)}))}_m.\end{array}$$

Consequently, one can unambiguously consider the free product of a finite number of quadratic subproduct systems.

Fusion Rules for the free product of quadratic subproduct systems

Let $\mathcal{H}={\{H_m\}}_{m\in {\mathbb{N}}_0}$ and $\mathcal{K}={\{K_m\}}_{m\in {\mathbb{N}}_0}$ be $r_0$-relator and $r_1$-relator generic quadratic subproduct systems in $d_0$ and $d_1$ generators, respectively, satisfying the condition in (8). Then we can apply the same argument from Remark 3.9 to deduce that their free product subproduct system also satisfies Condition (8) of having few relations, and hence it has Hilbert series

$$\begin{array}{c}\hfill h_{\mathcal{H}\star \mathcal{K}}(z)={(1-(d_0+d_1)z+(r_0+r_1)z^2)}^{-1}.\end{array}$$

Setting ${\delta }_m=dim({(H\star K)}_m)$, we obtain the sequence ${\{{\delta }_m\}}_{m\ge 0}$ satisfies the recurrence relation

$$\begin{array}{cc}\hfill {\delta }_0& =1;\hfill \\ \hfill {\delta }_1& =(d_0+d_1);\hfill \\ \hfill {\delta }_{m+1}& =(d_0+d_1)·{\delta }_m-(r_0+r_1)·{\delta }_{m-1},\hfill \end{array}$$

which implies

$$\begin{array}{c}\hfill {(H\star K)}_m\otimes {(H\star K)}_1={(H\star K)}_{m+1}\oplus {(H\star K)}_{m-1}^{\oplus (r_0+r_1)}.\end{array}$$ 12

Assuming that $\mathcal{H}$ and $\mathcal{K}$ are generic one-relator quadratic subproduct systems, the isomorphism (12) reduces to

$$\begin{array}{c}\hfill {(H\star K)}_m\otimes {(H\star K)}_1={(H\star K)}_{m+1}\oplus {(H\star K)}_{m-1}^{\oplus 2}\end{array}$$ 13

Fock Spaces of Free Products are Free Products of Fock Spaces

We start this subsection by discussing free products of Hilbert spaces. Our main references are [11, 12, 30].

In general, one can define the free product of a family of Hilbert spaces, but for the sake of readability, we shall focus here on the case of two Hilbert spaces only.

Definition 4.7

Let $(H_1,{\xi }_0)$ and $(H_2,{\xi }_0)$ be two Hilbert spaces with a distinguished normal vector ${\xi }_0$. Their free product is the space $(H,{\xi }_0)$ with

$$\begin{array}{c}\hfill H\simeq \mathbb{C}{\xi }_0\oplus \underset{p\ge 1}{⨁}\underset{i\in D_p}{⨁}(H_{i_1}^{\circ }\otimes \cdots \otimes H_{i_p}^{\circ }),\end{array}$$

where

$$\begin{array}{c}\hfill D_p=\{i=(i_1,i_2,\cdots ,i_p):i_j\in \{1,2\}\text{and}{i}_j\ne i_{j+1},\forall 1\le j\le p-1\},\end{array}$$

and $H_i^{\circ }$ is the orthocomplement of $\mathbb{C}{\xi }_0$ in $H_i$.

Applying Definition 4.7 to the Fock spaces of two quadratic subproduct systems, ${\mathcal{F}}_i:={\mathcal{F}}_{{\mathcal{H}}^{(i)}}$, with distinguished normal vector the vacuum vector ${\omega }_0$ for $H_0^{(1)}\simeq \mathbb{C}\simeq H_0^{(2)}$, we obtain

$$\begin{array}{c}\hfill {\mathcal{F}}_1\ast {\mathcal{F}}_2\simeq \mathbb{C}{\omega }_0\oplus \underset{p\ge 1}{⨁}\underset{i\in D_p}{⨁}({\mathcal{F}}_{i_1}^{+}\otimes \cdots \otimes {\mathcal{F}}_{i_p}^{+}),\end{array}$$

with the usual convention that ${\mathcal{F}}_i^{+}$ denotes the positive Fock space, i.e. $\mathcal{F}{({\mathcal{H}}^{(i)})}^{+}={⨁}_{n\ge 1}{H}_n^{(i)}$.

Proposition 4.8

Let $\mathcal{H}$ and $\mathcal{K}$ be quadratic subproduct systems of finite-dimensional Hilbert spaces. Then the Fock space $\mathcal{F}(\mathcal{H}\star \mathcal{K})$ is unitarily isomorphic to the Hilbert space free product of the Fock spaces $\mathcal{F}(\mathcal{H})\ast \mathcal{F}(\mathcal{K})$.

Proof

First, we consider the free product of the Fock spaces

$$\begin{array}{c}\hfill \mathcal{F}(\mathcal{H})\ast \mathcal{F}(\mathcal{K})=\mathbb{C}\oplus \underset{n\ge 1}{⨁}((\underset{n}{\underbrace{F{(\mathcal{H})}^{+}\otimes \mathcal{F}{(\mathcal{K})}^{+}\otimes \mathcal{F}{(\mathcal{H})}^{+}\otimes \cdots }})\oplus \end{array}$$ 14

$$\begin{array}{c}\hfill (\underset{n}{\underbrace{\mathcal{F}{(\mathcal{K})}^{+}\otimes \mathcal{F}{(\mathcal{H})}^{+}\otimes \mathcal{F}{(\mathcal{K})}^{+}\otimes \cdots }})).\end{array}$$ 15

Any direct summand of $\mathcal{F}(\mathcal{H})\ast \mathcal{F}(\mathcal{K})$ has the following form:

$$\begin{array}{c}\hfill H_{i_1}\otimes K_{j_1}\otimes H_{i_2}\otimes \cdots \otimes H_{i_l}\otimes K_{j_l},\end{array}$$ 16

where ${\sum }_{s=1}^l(i_s+j_s)=m$ for some $m\in \mathbb{N}$, $i_1\ge 0$, and $j_k\ge 1$, for all $1\le k\le l-1$. By Proposition 4.5, the vector space in (16) is a direct summand of ${(\mathcal{H}\star \mathcal{K})}_m$, which implies that $\mathcal{F}(\mathcal{H})\ast \mathcal{F}(\mathcal{K})\subset \mathcal{F}(\mathcal{H}\star \mathcal{K})$.

On the other hand, the reverse inclusion follows from the fact that each summand in ${(\mathcal{H}\star \mathcal{K})}_m$ is of the form $H_{i_1}\otimes K_{j_1}\otimes H_{i_2}\otimes \cdots \otimes H_{i_l}\otimes K_{j_l}$, with ${\sum }_{s=1}^l(i_s+j_s)=m$, hence a summand in the free product $\mathcal{F}(\mathcal{H})\ast \mathcal{F}(\mathcal{K})$. $\square$

By induction and by associativity of the operations of free product of Hilbert spaces and subproduct systems, the claim holds for finitely many quadratic subproduct systems:

Corollary 4.9

Let ${\mathcal{H}}^{(i)}$, for $i=0,\cdots .,n$, be $n+1$ quadratic subproduct systems of finite-dimensional Hilbert spaces. The Fock space $\mathcal{F}({\star }_{i=0}^n{\mathcal{H}}^{(i)})$ is unitarily isomorphic to the Hilbert space free product of the Fock spaces ${\ast }_{i=0}^n\mathcal{F}({\mathcal{H}}^{(i)})$.

Toeplitz Algebras and KK-Theory

In this section, we study the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}$ associated with the free product $\mathcal{H}\star \mathcal{K}$ of two quadratic subproduct systems $\mathcal{H}$ and $\mathcal{K}$, as per Definition  4.1.

Our main result for this section, Theorem 5.4, asserts that the Toeplitz algebra of such a free product is itself a free product, namely a reduced free product C*-algebra.

Free products of Toeplitz Algebras and Functoriality

Recall that the free product of algebras in (6) is the coproduct in the category of associative algebras over a field. For C$_{}^{\ast }$-algebras, there are minimal and maximal free products, making the question of which free product is the right categorical coproduct particularly relevant. As discussed in [7], if one considers GNS representations together with designated cyclic vectors, one can define a “free product representation” with a designated cyclic vector, thus obtaining what Avitzour [7] calls a small free product representation. Let us recall how the two constructions work and relate to each other.

Definition 5.1

Given two separable and unital C*-algebras $A_1$ and $A_2$, their unital full free product $A_1\star A_2$ is given by the following commuting diagram of one-to-one unital morphisms:

Definition 5.2

Let $\{(A_i,{\varphi }_i,{\mathcal{H}}_{{\varphi }_i},{\xi }_i):i=0,\cdots ,n\}$ be a family of unital $C^{\ast }$-algebras with GNS states, Hilbert spaces, and unit vectors. Let ${\lambda }_i$ denote the left multiplication. The reduced free product $(A,\varphi )={\ast }_i^r(A_i,{\varphi }_i)$ is the $C^{\ast }$-subalgebra generated by ${\cup }_{i\in I}{\lambda }_i(A_i)$, in the free product Hilbert space ${\star }_{i=0}^n({\mathcal{H}}_{{\varphi }_i},{\xi }_i)$.

When the GNS states are clear from the context, we simply write ${\ast }_{\mathbb{C}}{A}_i$ for the reduced free product of the algebras.

Consider now the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}}$ of a subproduct system of Hilbert spaces $\mathcal{H}$. By construction, ${\mathbb{T}}_{\mathcal{H}}$ acts on the Fock space $\mathcal{F}(\mathcal{H})={⨁}_{m\ge 0}{H}_m$ faithfully via the left shift operators. We denote this $\ast$-representation by $({\mathcal{F}}_{\mathcal{H}},\tau )$ and refer to it as the Toeplitz representation (cf. [32, Definition 2.13]).

Proposition 5.3

The Toeplitz representation $({\mathcal{F}}_{\mathcal{H}},\tau )$ is equivalent to the GNS representation induced by the state $\phi :{\mathbb{T}}_{\mathcal{H}}\to \mathbb{C}$ given by $\phi (T):=⟨T({\omega }_0),{\omega }_0⟩$, with ${\omega }_0$ the unit vector in $H_0$.

Note that the state $\phi$ is the projection from ${\mathbb{T}}_{\mathcal{H}}$ onto the complex numbers. This fact, combined with the discussion in the previous section, yields:

Theorem 5.4

Let $\mathcal{H}$ and $\mathcal{K}$ be quadratic subproduct systems of finite-dimensional Hilbert spaces, and let $\mathcal{H}\star \mathcal{K}$ be the free product of the subproduct systems in Definition 4.1. We have

$$\begin{array}{c}\hfill {\mathbb{T}}_{\mathcal{H}}{\ast }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}}\cong {\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}.\end{array}$$

We shall now elaborate on the categorical aspects of our construction. Recall that the category ${\text{Hilb}}_f$ of finite-dimensional Hilbert spaces is the primary example of a strict C$_{}^{\ast }$-tensor category, with morphisms being linear maps and the unit object being the uniquely defined one-dimensional Hilbert space $\mathbb{C}$. In particular, we consider the subcategory ${\text{Hilb}}_f^1$ where morphisms are isometric maps.

Recall that a lax monoidal functor [20, Section XII.2] is a functor between two monoidal categories, together with two coherence maps satisfying associativity and unitality.

Definition 5.5

A subproduct system is a lax monoidal functor from $({\mathbb{N}}_0,+,0)$ to ${\text{Hilb}}_f^1$. The category of subproduct systems of finite-dimensional Hilbert spaces ${\text{SPS}}_f^{\mathbb{C}}$ is the subcategory of ${({\text{Hilb}}_f^1)}^{{\mathbb{N}}_0}$ whose objects are lax monoidal functors from ${\mathbb{N}}_0$ to ${\text{Hilb}}_f^1$ and whose morphisms are as in [29, Definition  1.4].

Let us now restrict to the subcategory ${\text{SPS}}_{f,2}^{\mathbb{C}}$ of quadratic subproduct systems of finite-dimensional Hilbert spaces. Then there is a functor $\mathbf{Toe}$ from the category ${\text{SPS}}_{f,2}^{\mathbb{C}}$ to the category of unital separable C*- algebras with states, that associates every quadratic subproduct system of finite-dimensional Hilbert spaces $\mathcal{H}$ with the corresponding Toeplitz C*-algebra and distinguished state $({\mathbb{T}}_{\mathcal{H}},\tau )$ as in as in Proposition 5.3, and to every morphism of subproduct systems, the corresponding *-homomorphism at the level of C$_{}^{\ast }$-algebras.

Our construction of the free product of quadratic subproduct systems is therefore natural, as it is mapped to the corresponding free product of C$_{}^{\ast }$-algebras by the functor $\mathbf{Toe}$.

Example 5.6

The Cuntz algebras ${\mathcal{O}}_n$ can be realised as quotients of the n-fold free product of Toeplitz algebras $\mathbb{T}$, which in turn can be realised as Toeplitz algebras of the (sub)product system with $H_n:=\mathbb{C}$ for all n.

Example 5.7

(Cuntz–Krieger algebras and quadratic monomial ideals) Monomial ideals are a special class of ideals, and subproduct systems associated with monomial ideals [17] give rise to many well-studied operator algebras, including Cuntz–-Krieger and subshift C*-algebras á la Matsumoto [23].

Let us recall how Cuntz–Krieger algebras can be described using subproduct systems. Let $A\in {\text{Mat}}_n\{0,1\}$, with no row or column equal to zero. Inside the free algebra $\mathbb{C}⟨X_1,\cdots ,X_n⟩$ we consider the quadratic monomial ideal generated by the quadratic monomials corresponding to the zero entries of the matrix A, i.e.,

$$\begin{array}{c}\hfill J_A:=⟨X_i{X}_j:1\le i,j\le n,A_{\mathit{ij}}=0⟩.\end{array}$$

The corresponding subproduct system ${\mathcal{H}}^A$ has fibres $H_m^A$ spanned by the admissible words of length m in the alphabet $\{X_1,\cdots ,X_n\}$ and agrees with the subproduct system of a Matsumoto shift in the sense of [29]. Note that A can be interpreted as the incidence matrix of the underlying Markov chain.

Let now $B\in {\text{Mat}}_m\{0,1\}$, satisfying again the condition of having no row or column equal to zero. Consider the associated subproduct system ${\mathcal{H}}^B$ and take the free product subproduct system ${\mathcal{H}}^A\star {\mathcal{H}}^B$. It is easy to see that this is the subproduct system of the Cuntz–Krieger algebra of the $(n+m)\times (n+m)$ matrix

$$\begin{array}{c}\hfill \left(\begin{array}{ccc}A& |& \mathbf{1}\\ \mathbf{1}& |& B\end{array}\right),\end{array}$$

where $\mathbf{1}$ denotes the matrix with all entries equal to one.

To describe what this means at the level of the underlying Markov chains, we need the following standard notion from graph theory.

Definition 5.8

([16, Page 21]) Given two directed graphs $E=(E^0,E^1),F=(F^0,F^1)$, their join $E+F$ is the graph with vertex set

$$\begin{array}{c}\hfill {(E+F)}^0:=E^0\cup E^1,\end{array}$$

and edge set

$$\begin{array}{c}\hfill {(E+F)}^1:=E^1\cup F^1\cup \left\{(v_1,v_2),,,(v_2,v_1),:,\forall ,v_1,\in ,E^0,,,v_2,\in ,F^0\right\},\end{array}$$

where $(v_1,v_2)$ denotes the arrow from $v_1$ to $v_2$.

Given matrices A and B with underlying Markov chains $E_A$ and $E_B$, the free product of their subproduct systems is the quadratic subproduct with underlying Markov chain their graph join $E_A+E_B$. Correspondingly, the Cuntz–Pimsner algebra is the Cuntz–Krieger algebra of the incidence matrix of the graph join of the two underlying directed graphs.

The examples above can also be interpreted as special cases of a result on Cuntz–Pimsner algebras of correspondences, due to Speicher [30] (see also [9]), stating that the Toeplitz algebra of the finite direct sum of C*-correspondences over the same coefficient algebra A is the amalgamated free product over A of the Cuntz–Pimsner algebras of a single correspondence.

Theorem 5.9

([9, Example 4.7.5]) Let $H_i$, be a family of C*-correspondences over A. Denote the corresponding Toeplitz–Pimsner algebras by $\mathcal{T}(H_i)$, and by $E_{H_i}$ their conditional expectations. We have

$$\begin{array}{c}\hfill \left(\mathbb{T},({\oplus }_i{H}_i),,,E_{\oplus H_i}\right)\simeq {\star }_i^r(\mathbb{T}(H_i),E_{H_i}).\end{array}$$

In the next section, we will focus on free products of a special class of proper subproduct systems. To our knowledge, this is the first time that such examples have been studied. However, before doing that, we shall first discuss the KK-theory and nuclearity of our Toeplitz algebras.

KK-Theory and Nuclearity for Free Products

We shall now recall some known results about the K-theory of free products of C$_{}^{\ast }$-algebras. Our main references are [7, 11, 12].

For nuclear $C^{\ast }$-algebras $A_1$ and $A_2$ and any separable $C^{\ast }$-algebra E, Germain has proven that the reduced free product $A_1{\star }_{\mathbb{C}}{A}_2$ is KK-equivalent to the unital full free product $A_1\star A_2$. This implies the existence of the following six-term exact sequence:

We will use (17) to compute the K-theory groups of the Toeplitz algebra of the reduced free product of two quadratic subproduct systems.

Definition 5.10

([11, Definition 5.1]) A unital $C^{\ast }$-algebra A is said to be K-pointed if there exists $\alpha \in KK(A,\mathbb{C})$ such that $i_A^{\ast }(\alpha )=1_{\mathbb{C}}$ with $i_A$ the inclusion of $\mathbb{C}$ in A given by the unit.

Observe that if A is a unital $C^{\ast }$-algebra and KK-equivalent to $\mathbb{C}$, then A is K-pointed.

Theorem 5.11

([11, Theorem 5.5]) Let $A_0$ and $A_1$ be two K-pointed $C^{\ast }$-algebras. Then $A_0\oplus A_1$ is KK-equivalent to $A_0\star A_1\oplus \mathbb{C}$.

Corollary 5.12

Let $A_0$ and $A_1$ be two K-pointed $C^{\ast }$-algebras. If $A_0$ and $A_1$ belong to the UCT class $\mathcal{N}$, then their reduced free product $A_0{\star }_{\mathbb{C}}{A}_1$ also belongs to $\mathcal{N}$.

Proof

The UCT class $\mathcal{N}$ is closed under direct sum and KK-equivalence. This fact, combined with Theorem 5.11, yields $A_0\oplus A_1{\sim }_{\mathit{KK}}{A}_0\star A_1\oplus \mathbb{C}\in \mathcal{N}$. Furthermore, since $A_0\star A_1$ is K-dominated by $A_0\star A_1\oplus \mathbb{C}$, the claim follows.$\square$

Theorem 5.13

Let $\mathcal{H}$ and $\mathcal{K}$ be quadratic; subproduct systems of finite-dimensional Hilbert spaces. Assume that the Toeplitz algebras ${\mathbb{T}}_{\mathcal{H}}$ and ${\mathbb{T}}_{\mathcal{K}}$ are both nuclear and KK-equivalent to the complex numbers. Then so is the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}$.

Proof

By Corollary 5.12, under the assumptions of the theorem, it follows that ${\mathbb{T}}_H{\star }_{\mathbb{C}}{\mathbb{T}}_K\in \mathcal{N}$. Consequently, it suffices to show that ${\mathbb{T}}_{\mathcal{H}}\star {\mathbb{T}}_{\mathcal{K}}$ has the same K-theory as $\mathbb{C}$, thanks to the fact that ${\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}\cong {\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}}$. After replacing E with $\mathbb{C}$ and $A_0,A_1$ with ${\mathbb{T}}_{\mathcal{H}}$ and ${\mathbb{T}}_{\mathcal{K}}$, respectively, the long exact sequence (17) becomes

$$0⟶K_1({\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}})⟶\mathbb{Z}\stackrel{(i_1+i_2)_{\ast }^{}}{\to }\mathbb{Z}\oplus \mathbb{Z}⟶K_0({\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}})⟶0.$$

From this, we compute that

$$\begin{array}{cc}& K_1({\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}})\cong ker(i_1+i_2)_{\ast }^{}\cong \{0\}\cong K_1(\mathbb{C}),\hfill \\ \hfill & K_0({\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}})\cong \text{coker}(i_1+i_2)_{\ast }^{}\cong \mathbb{Z}\cong K_0(\mathbb{C}),\hfill \end{array}$$

which implies that ${\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}}$ is KK-equivalent to $\mathbb{C}$. By the definition of free product of subproduct systems and functoriality, we have ${\mathbb{T}}_{\mathcal{H}}{\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}}\cong {\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}$. Therefore, ${\mathbb{T}}_{\mathcal{H}\star \mathcal{K}}$ is KK-equivalent to $\mathbb{C}$.

By assumption, ${\mathbb{T}}_{\mathcal{H}},{\mathbb{T}}_{\mathcal{K}}$ are nuclear and KK-equivalent to $\mathbb{C}$. Moreover, the compact operators $\mathbb{K}(\mathcal{F})$ are contained in the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}}$ (see. [33, Corollary 3.2]). Thus, thanks to [26, Theorem 1.1], we obtain nuclearity of the reduced free product ${\star }_{\mathbb{C}}{\mathbb{T}}_{\mathcal{K}}$. $\square$

A Case Study: Free Products of Temperley–Lieb Subproduct Systems

Definition 6.1

([14, Definition 1.2] Let H be a finite-dimensional Hilbert space of dimension $m\ge 2$. A non-zero vector $P\in H\otimes H$ is called Temperley–Lieb if there is $\lambda >0$ such that the orthogonal projection $e:H\otimes H\to \mathbb{C}·P$ satisfies

$$\begin{array}{c}\hfill (e\otimes 1)(1\otimes e)(e\otimes 1)={\displaystyle \frac{1}{\lambda }}(e\otimes 1)\text{in}B(H\otimes H\otimes H).\end{array}$$

We will often fix an orthonormal basis in H and identify $H^{\otimes n}$ with the space of homogeneous noncommutative polynomials of degree n in variables $X_1,\cdots ,X_d$. In particular, we write a vector $P\in H\otimes H$ as a noncommutative polynomial $P={\sum }_{i,j=1}^d{a}_{\mathit{ij}}{X}_i{X}_j$. Consider the matrix $A={(a_{\mathit{ij}})}_{i,j}$. By [14, Lemma 1.4], P is Temperley–Lieb if and only if the matrix $A\overline{A}$ is unitary up to a (non-zero) scalar factor, where $\overline{A}={({\overline{a}}_{\mathit{ij}})}_{i,j}$. Since the ideal generated by P does not change if we multiply P by a non-zero factor, we may always assume that $A\overline{A}$ is unitary.

The standard subproduct system ${\mathcal{H}}^P$ defined by the ideal $⟨P⟩\subset T(H)$ generated by P is called a Temperley–Lieb subproduct system. We write ${\mathcal{F}}_P={\mathcal{F}}_{{\mathcal{H}}_P}$, ${\mathbb{T}}_P={\mathbb{T}}_{{\mathcal{H}}_P}$ and ${\mathbb{O}}_P={\mathbb{O}}_{{\mathcal{H}}_P}$.

The following result gives a complete set of relations in ${\mathbb{T}}_P$.

Theorem 6.2

([15, Theorem 2.11]) Let $A={(a_{\mathit{ij}})}_{i,j}\in {\text{GL}}_d(\mathbb{C})$ ($m\ge 2$) be such that $A\overline{A}$ is unitary. Let $q\in (0,1]$ be the number such that $\text{Tr}(A^{\ast }A)=q+q^{-1}$. Consider the noncommutative polynomial $P={\sum }_{i,j=1}^d{a}_{\mathit{ij}}{X}_i{X}_j$. Then ${\mathbb{T}}_P$ is the universal C$_{}^{\ast }$-algebra generated by the C$_{}^{\ast }$–algebra $c:=C({\mathbb{Z}}_{+}\cup \{\infty \})$ and elements $S_1,S_2,...,S_d$ satisfying the relations

$$\begin{array}{c}\hfill f{S}_i=S_i\gamma (f)(f\in c,1\le i\le d),\sum _{i=1}^d{S}_i{S}_i^{\ast }=1-e_0,\sum _{i,j=1}^d{a}_{\mathit{ij}}{S}_i{S}_j=0,\end{array}$$

$$\begin{array}{c}\hfill S_i^{\ast }{S}_j+\varphi \sum _{k,l=1}^d{a}_{\mathit{ik}}{\overline{a}}_{\mathit{jl}}{S}_k{S}_l^{\ast }={\delta }_{\mathit{ij}}1(1\le i,j\le d),\end{array}$$

where $\gamma :c\to c$ is the shift to the left (so $\gamma (f)(n)=f(n+1)$), $e_0$ is the characteristic function of $\{0\}$ and $\varphi \in c$ is the element given by

$$\begin{array}{c}\hfill \varphi (n)={\displaystyle \frac{{[n]}_q}{{[n+1]}_q}},\text{with}{[n]}_q={\displaystyle \frac{q^n-q^{-n}}{q-q^{-1}}}.\end{array}$$ 18

Here c is identified with a unital subalgebra of $\mathbb{K}({\mathcal{F}}_P)+\mathbb{C}1\subset {\mathbb{T}}_P$, with $e_n\in c$ being identified with the projection ${\mathcal{F}}_P\to H_n$. Note also that $\varphi (n)\to q$ as $n\to +\infty$.

The relations become slightly simpler if we write P in a standard form. Namely, by [14, Proposition 1.5], up to a unitary change of variables and rescaling, we may assume that our Temperley–Lieb polynomial P has the form

$$\begin{array}{c}\hfill P=\sum _{i=1}^m{a}_i{X}_i{X}_{m-i+1},\text{with}|a_i{a}_{m-i+1}|=1.\end{array}$$ 19

Fusion rules for free products of Temperley–Lieb subproduct systems

For $i=1,2$, let $P_i$ be Temperley–Lieb polynomials, with ${\mathcal{H}}^{P_i}$ their corresponding Temperley–Lieb subproduct systems. For ease of notation, denote their free product ${\mathcal{H}}^{⟨P_1,P_2⟩}$ by ${\mathcal{H}}^{\star }$. In particular, we have

$$\begin{array}{cc}\hfill H_0^{\star }& \cong \mathbb{C};\hfill \\ \hfill H_1^{\star }& \cong H_1^1\oplus H_1^2\cong {\text{Span}}_{\mathbb{C}}\{e_1^1,\cdots ,e_{d_1}^1,e_1^2,\cdots ,e_{d_2}^2\};\hfill \\ \hfill H_2^{\star }& \cong H_1^{\star }\otimes H_1^{\star }\ominus {\text{Span}}_{\mathbb{C}}\left\{P_1,(e_1^1,\cdots ,e_{d_1}^1),,,P_2,(e_1^2,\cdots ,e_{d_2}^2)\right\},\hfill \end{array}$$

where $d_i=dim(H_1^i)$, and $\{e_j^i:j=1,2,\cdots ,d_i\}$ forms an orthonormal basis of $H_1^i$.

In [15], the author constructed maps

$$\begin{array}{c}\hfill w_n^P:={({[2]}_q\varphi (n+1))}^{\frac{1}{2}}(f_{n+1}\otimes 1)(1^{\otimes n}\otimes v):H_n^P\to H_{n+1}^P\otimes H_1^P,\end{array}$$

where $v=P(e)\in H_1^P\otimes H_1^P$ is the Temperley–Lieb vector corresponding to the polynomial P.

We should use a small adaptation of those maps to construct an explicit isometry

$$\begin{array}{c}\hfill w_n^{\star }:{(H_{n-1}^{\star })}^{\oplus 2}\oplus H_{n+1}^{\star }\to H_n^{\star }\otimes H_1^{\star }.\end{array}$$

For $i=1,2$, consider the two projections $Q_i:H_1^{\star }\cong H_1^1\oplus H_1^2$ onto $H_1^1$ and $H_1^2$, respectively. Define the map

$$\begin{array}{c}\hfill \begin{array}{cc}& w_n^i={({[2]}_q)}^{1/2}({\varphi }^i{(n+1))}^{\frac{1}{2}}{Q}_i^{\otimes n+2}(f_{n+1}\otimes 1)(1^{\otimes n}\otimes v_i)Q_i^{\otimes n}\hfill \\ \hfill & +{({\varphi }^i(n))}^{\frac{1}{2}}(1-Q_i)\otimes Q_i^{\otimes n+1}(1\otimes f_n\otimes 1)(1\otimes 1^{\otimes n-1}\otimes v_i)(1-Q_i)\otimes Q_i^{\otimes n-1}\hfill \\ \hfill & +{({\varphi }^i(n-1))}^{\frac{1}{2}}1\otimes (1-Q_i)\otimes Q_i^{\otimes n}(1^{\otimes 2}\otimes f_{n-1}\otimes 1)(1^{\otimes n}\otimes v_i)1\otimes (1-Q_i)\otimes Q_i^{\otimes n-2}\hfill \\ \hfill & +\cdots \hfill \\ \hfill & +{({\varphi }^i(1))}^{\frac{1}{2}}{1}^{\otimes n-1}\otimes (1-Q_i)\otimes Q_i^{\otimes 2}(1^{\otimes n}\otimes f_1\otimes 1)(1^{\otimes n}\otimes v_i)1^{\otimes n-1}\otimes (1-Q_i)).\hfill \end{array}\end{array}$$ 20

Proposition 6.3

The map $w_n^{\star }:=w_n^1\oplus w_n^2:{(H_{n-1}^{\star })}^{\oplus 2}\to H_n^{\star }\otimes H_1^{\star }$ is an isometry.

Combining this with the structure map ${\iota }_{n,1}^{\star }:H_{n+1}^{\star }\mapsto H_n^{\star }\otimes H_1^{\star }$ we obtain the following:

Theorem 6.4

There is a unitary isomorphism:

$$\begin{array}{c}\hfill W_n^R:=(w_n^{\star },{\iota }_{n,1}^{\star }):{(H_{n-1}^{\star })}^{\oplus 2}\oplus H_{n+1}^{\star }\to H_n^{\star }\otimes H_1^{\star }\end{array}$$

Proof

Let $v_i=P_i(e)\in H_1^{(i)}\otimes H_1^{(i)}$ be the Temperley–Lieb vector corresponding to the polynomial $P_i$. Using (13), it is not hard to see that $w_n^{\star }$ is orthogonal to ${\iota }_{n,1}^{\star }$.

Note that, in the image of the map $w_n^i$, the last component belongs to $H_1^{(i)}$, one of the orthogonal components. This yields

$$\begin{array}{c}\hfill ⟨w_n^i({\xi }_1),w_n^j({\xi }_2)⟩=0,\text{for all}{\xi }_1,{\xi }_2\in H_{n-1}^{\star },i\ne j.\end{array}$$

It remains to prove that each $w_n^i$ is an isometry. Let $\xi \in H_n^{\star }$. It is not hard to see that the summands of $w_n^i(\xi )$ are mutually orthogonal, since the images of $Q_i$ and $1-Q_i$ are.

For any vector $\xi \in \text{Im}(1^{\otimes k}\otimes (1-Q_i)\otimes Q_i^{\otimes n-k+1})$, we have that $⟨w_n^k(\xi ),w_n^k(\xi )⟩$ is equal to

$$\begin{array}{c}\hfill {∥{({[2]}_q{\varphi }^i(n-k))}^{\frac{1}{2}},(1^{\otimes k}\otimes f_{n-k+1}\otimes 1),(1^{\otimes k}\otimes 1^{\otimes n-k}\otimes v_i),(\xi )∥}^2.\end{array}$$

Write $(1^{\otimes k}\otimes f_{n-k+1}\otimes 1)(1^{\otimes k}\otimes 1^{\otimes n-k}\otimes v_i)$ as $1^{\otimes k}\otimes (f_{n-k+1}\otimes 1)(1^{\otimes n-k}\otimes v_i)$, the tensor product of linear operators. Let $\{{\eta }_j:j=1,2,\cdots ,dim{(H_1^{\star })}^k\}$ be the basis of ${H_1^{\star }}^{\otimes k}$ and write $\xi ={\sum }_{j=1}^{dim{(H_1^{\star })}^k}{\eta }_j\otimes {\zeta }_j$ for some ${\zeta }_j\in H_{n-k}^{\star }$. Then we have

$$\begin{array}{cc}& {∥{({[2]}_q{\varphi }^i(n-k))}^{\frac{1}{2}},(1^{\otimes k}\otimes f_{n-k+1}\otimes 1),(1^{\otimes k}\otimes 1^{\otimes n-k}\otimes v_i),(\xi )∥}^2\hfill \\ \hfill & ={∥{({[2]}_q{\varphi }^i(n-k))}^{\frac{1}{2}},1^{\otimes k},\otimes ,(f_{n-k+1}\otimes 1),(1^{\otimes n-k}\otimes v_i),\left(\sum _{j=1}^{dim{(H_1^{\star })}^k},{\eta }_i,\otimes ,{\zeta }_i\right)∥}^2\hfill \\ \hfill & ={∥\sum _{j=1}^{dim{(H_1^{\star })}^k},{\eta }_i,\otimes ,{({[2]}_q{\varphi }^i(n-k))}^{\frac{1}{2}},·,(f_{n-k+1}\otimes 1),(1^{\otimes n-k}\otimes v_i),({\zeta }_i)∥}^2\hfill \\ \hfill & =\sum _{j=1}^{dim{(H_1^{\star })}^k}\Vert {\zeta }_j{\Vert }^2={\Vert \xi \Vert }^2,\hfill \end{array}$$

where the second to last equality follows from [15, Equation 4.1]. This proves the claim. $\square$

For ease of notation, we write

$$\begin{array}{cc}& V_R:={\oplus }_{n=1}^{\infty }{W}_n^R:{\oplus }_{n=0}^{\infty }{(H_n^{\star })}^{\oplus 2}\cong {\mathcal{F}}^{\oplus 2}\to {\oplus }_{n=1}^{\infty }{H}_n^{\star }\otimes H_1^{\star }\cong {\mathcal{F}}_{+}\otimes H_1^{\star },\hfill \\ \hfill & w^i={\oplus }_{n=0}^{\infty }{w}_n^i:{\oplus }_{n=0}^{\infty }{H}_n^{\star }\cong \mathcal{F}\to {\oplus }_{n=1}^{\infty }{H}_n^{\star }\otimes H_1^{\star }\cong {\mathcal{F}}_{+}\otimes H_1^{\star },\text{for i = 1, 2.}\hfill \end{array}$$

These two maps are important in the construction of a $KK(\mathbb{T},\mathbb{C})$ element in Section 6.2.1.

Example 6.5

Consider the Temperley–Lieb polynomials

$$\begin{array}{c}\hfill P_1(X_1,X_2)=X_1{X}_2-X_2{X}_1,\end{array}$$ 21

$$\begin{array}{c}\hfill P_2(X_3,X_4,X_5)=a_1·X_3{X}_5+a_2·X_4{X}_4+a_3·X_5{X}_3.\end{array}$$ 22

such that $|a_1{|}^2+|a_2{|}^2+{|a_3|}^2=q+q^{-1}$ for $q\in (0,1]$. The Temperley–Lieb subproduct systems induced by $P_1$ and $P_2$ are the maximal subproduct systems with fibres

$$\begin{array}{cc}\hfill H_1^1& ={\text{Span}}_{\mathbb{C}}\{e_1,e_2\},\hfill \\ \hfill H_2^1& ={\text{Span}}_{\mathbb{C}}{\{e_1\otimes e_2-e_2\otimes e_1\}}^{\perp },\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_1^2& ={\text{Span}}_{\mathbb{C}}\{e_3,e_4,e_5\},\hfill \\ \hfill H_2^2& ={\text{Span}}_{\mathbb{C}}{\{a_1·e_3\otimes e_5+a_2·e_4\otimes e_4+a_3·e_5\otimes e_3\}}^{\perp },\hfill \end{array}\end{array}$$

respectively.

The free product subproduct system has fibres

$$\begin{array}{cc}\hfill H_1^{⟨P_1,P_2⟩}& =H_1^1\oplus H_1^2,\hfill \\ \hfill H_2^{⟨P_1,P_2⟩}& ={\text{Span}}_{\mathbb{C}}{\{e_1\otimes e_2-e_2\otimes e_1,a_1·e_3\otimes e_5+a_2·e_4\otimes e_4+a_3·e_5\otimes e_3\}}^{\perp },\hfill \end{array}$$

and satisfies (11). Consider the case when $n=1$, we have the decomposition:

$$\begin{array}{c}\hfill (w_1^1,w_1^2):H_1^{⟨P_1,P_2⟩}\oplus H_1^{⟨P_1,P_2⟩}\to H_2^{⟨P_1,P_2⟩}\otimes H_1^{⟨P_1,P_2⟩}\end{array}$$

Applying $w_1^2$ defined in (20), we obtain:

$$\begin{array}{cc}& w_1^2(e_1)=e_1\otimes v_2,\hfill \\ \hfill & w_1^2(e_2)=e_2\otimes v_2,\hfill \\ \hfill & w_1^2(e_3)=\frac{{[2]}_q}{{[3]}_q^{1/2}}·\left(e_3,\otimes ,v_2,-,\frac{a_1{a}_3}{q+q^{-1}},·,v_2,\otimes ,e_3\right),\hfill \\ \hfill & w_1^2(e_4)={\left(\frac{{[2]}_q^2}{{[3]}_q}\right)}^{\frac{1}{2}}·\left(e_4,\otimes ,v_2,-,\frac{a_2^2}{q+q^{-1}},·,v_2,\otimes ,e_4\right),\hfill \\ \hfill & w_1^2(e_5)={\left(\frac{{[2]}_q^2}{{[3]}_q}\right)}^{\frac{1}{2}}·\left(e_5,\otimes ,v_2,-,\frac{a_1{a}_3}{q+q^{-1}},·,v_2,\otimes ,e_5\right),\hfill \end{array}$$

where $v_2:=a_1·e_3\otimes e_5+a_2·e_4\otimes e_4+a_3·e_5\otimes e_3$ is the quadratic relation that defines Temperley–Lieb subproduct system ${\mathcal{H}}^{P_2}$.

By associativity, the above construction can be extended to the free product of a finite number of Temperley–Lieb subproduct systems:

Theorem 6.6

Let $P_i,i=1,\cdots ,r$ be r Temperley–Lieb polynomials with associated Temperley–Lieb subproduct systems ${\mathcal{H}}^{P_i}$. Denote by ${\mathcal{H}}^{\star }$ their free product. There is a unitary isomorphism

$$W_n^R:=(w^{\star },{\iota }_{n,1}^{\star }):{(H_{n-1}^{\star })}^{\oplus r}\oplus H_{n+1}^{\star }\to H_n^{\star }\otimes H_1^{\star },$$

where ${\iota }_{n,1}^{\star }$ are the structure maps of ${\mathcal{H}}^{\star }$, and $w^{\star }:=(w_n^1,w_n^2,\cdots ,w_n^r)$ with $w_n^i$ as in (20).

Gysin Sequences

In this section, we will construct a noncommutative Gysin sequence for the Toeplitz algebra of the free product of Temperley–Lieb subproduct systems. This will allow us to simplify the six-term exact sequence in K-theory induced by the extension (1).

We start by recalling what is known about the K-theory of the Toeplitz and Cuntz–Pimsner algebras of a subproduct system.

Theorem 6.7

([15, Theorem 3.1 and Corollary 4.4]) For a Temperley–Lieb polynomial P, the inclusion $i:\mathbb{C}\to {\mathbb{T}}_P$ is a KK-equivalence. Moreover, we have $[e_0]=(2-m)[1]$ in $K_0({\mathbb{T}}_P)$.

As a consequence, the six-term exact sequence induced by the defining extension (1) simplifies notably, and one obtains the following result about the K-theory of the Cuntz–Pimsner algebra of a Temperley–Lieb subproduct system.

Corollary 6.8

([15, Corollary 4.4]) For every Temperley–Lieb polynomial P in d variables,

$$K_0({\mathbb{O}}_P)\cong \mathbb{Z}/(d-2)\mathbb{Z},K_1({\mathbb{O}}_P)\cong \left\{\begin{array}{cc}\mathbb{Z},\hfill & d=2,\hfill \\ 0,\hfill & d\ge 3.\hfill \end{array}\right)$$

Given that the Toeplitz algebras associated with Temperley–Lieb subproduct systems satisfy the conditions of Theorem 5.13, we derive the following result:

Theorem 6.9

Let $P_1$ and $P_2$ be Temperley–Lieb polynomials, with Toeplitz algebras ${\mathbb{T}}_{P_1}$ and ${\mathbb{T}}_{P_2}$. Then the Toeplitz algebra of the free product subproduct system ${\mathbb{T}}_{⟨P_1,P_2⟩}$ is isomorphic to the reduced free product ${\mathbb{T}}_{P_1}{\star }_{\mathbb{C}}{\mathbb{T}}_{P_2}$ and it is KK-equivalent to the algebra of complex numbers $\mathbb{C}$.

An Explicit KK-Equivalence

We shall now make our KK-equivalence result more explicit. To do so, we shall employ arguments similar to those in [5, 14].

With the same notation as in the previous section, let $P_1$ and $P_2$ be two Temperley–Lieb polynomials with associated subproduct systems ${\mathcal{H}}^{P_1}$ and ${\mathcal{H}}^{P_2}$, and let us consider their free product by ${\mathcal{H}}^{P_1}\star {\mathcal{H}}^{P_2}={\mathcal{H}}^{⟨P_1,P_2⟩}.$ In what follows, we will omit the subscript $⟨P_1,P_2⟩$ and write $\mathbb{T}$ for ${\mathbb{T}}_{⟨P_1,P_2⟩}$ and $\mathcal{F}$ for ${\mathcal{F}}_{⟨P_1,P_2⟩}$.

By Theorem 6.4 we have maps $W_n^R:H_n\otimes H_1\to H_{n+1}\oplus H_{n-1}^{\oplus 2}$, for every n. This allows us to construct a map $W_R:={(\iota ,V_R)}^{\ast }:\mathcal{F}\otimes H_1\to {\mathcal{F}}^{\oplus 3}$, with range ${\mathcal{F}}_{+}\oplus {\mathcal{F}}^{\oplus 2}$.

We consider the pair of homomorphisms $({\psi }_{+},{\psi }_{-})$, where ${\psi }_{\pm }:\mathbb{T}\to \mathcal{L}({\mathcal{F}}^{\oplus 3})$ are given by

$$\begin{array}{cc}& {\psi }_{+}(x)=x^{\oplus 3},\hfill \\ \hfill & {\psi }_{-}(x)=W_R(x\otimes 1_{H_1})W_R^{\ast }.\hfill \end{array}$$

We will show that the above pair $({\psi }_{+},{\psi }_{-})$ gives a KK-element which is a left and right inverse to the KK-class of the inclusion $i:\mathbb{C}\to \mathbb{T}$.

Lemma 6.10

The pair $({\psi }_{+},{\psi }_{-})$ defines an element $[{\psi }_{-},{\psi }_{+}]$ in $KK(\mathbb{T},\mathbb{C})$.

Proof

It is sufficient to prove that for all $x\in \mathbb{T}$, ${\psi }_{+}(x)-{\psi }_{-}(x)\in \mathbb{K}({\mathcal{F}}^{\oplus 3})$.

Let $d_1=dim(H_1)$. Since the Toeplitz algebra $\mathbb{T}$ is generated by the Toeplitz operators $T_i^{\ast }:=T_{e_i}^{\ast }$, where $\{e_i:i=1,2,\cdots ,d_1\}$ is an orthonormal basis of $H_1$, we only need to show that ${\psi }_{+}(T_i^{\ast })-{\psi }_{-}(T_i^{\ast })$ is a compact operator for all i.

Writing $W_R(x\otimes 1_{H_1})W_R^{\ast }$ in matrix form, we are left with checking that

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}{T}_i^{\ast }-{\iota }^{\ast }(T_i^{\ast }\otimes 1)\iota & {\iota }^{\ast }(T_i^{\ast }\otimes 1)w^1& {\iota }^{\ast }(T_i^{\ast }\otimes 1)w^2\\ {(w^1)}^{\ast }(T_i^{\ast }\otimes 1)\iota & T_i^{\ast }-{(w^1)}^{\ast }(T_i^{\ast }\otimes 1)w^1& {(w^1)}^{\ast }(T_i^{\ast }\otimes 1)w^2\\ {(w^2)}^{\ast }(T_i^{\ast }\otimes 1)\iota & {(w^2)}^{\ast }(T_i^{\ast }\otimes 1)w^1& T_i^{\ast }-{(w^2)}^{\ast }(T_i^{\ast }\otimes 1)w^2\end{array}\right]\\ \hfill =0mod\mathbb{K}({\mathcal{F}}^{\oplus 3}).\end{array}$$

Since $\iota$ is the inclusion of $H_{n+1}$ into $H_n\otimes H_1$ and $T_i^{\ast }$ commutes with the structure maps of the subproduct system, we have

$$(T_i^{\ast }\otimes 1)\iota (\xi )=T_i^{\ast }({\xi }_k)\otimes e_k=\iota (T_i^{\ast }\otimes 1)$$

for all $\xi \in H_{n+1}\subset H_n\otimes H_1$, where we write $\xi ={\sum }_k{\xi }_k\otimes e_k$. Consequently,

$$\begin{array}{c}\hfill (T_i^{\ast }\otimes 1)\iota =\iota (T_i^{\ast }\otimes 1).\end{array}$$

Moreover, ${(w^i)}^{\ast }\iota =0$, so the lower triangular part of ${\psi }_{+}(T_i^{\ast })-{\psi }_{-}(T_i^{\ast })$ vanishes.

To finalize the proof, we need to show that $(T_i^{\ast }\otimes 1)w^k=w^k{T}_i^{\ast }mod\mathbb{K}$, which together with the fact that $w^k$ is an isometry for all $k=1,2$, gives the desired result that $({\psi }_{+}-{\psi }_{-})(T_i^{\ast })=0$ modulo compact operators.

Let $\xi \in H_n$, using the fact that

$$\begin{array}{c}\hfill ((T_i^{\ast }\otimes 1)w^k-w^k{T}_i^{\ast })(\xi )=T_i^{\ast }(w_n-1\otimes w_{n-1})(\xi )\end{array}$$

as in the proof of [15, Lemma 4.1], it suffices to prove that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\Vert (w_n^k-1\otimes w_{n-1}^k)f_n\Vert =0,\text{for all}k=1,2.\end{array}$$

To this end, we observe that

$$\begin{array}{cc}& {((w_n^k-1\otimes w_{n-1}^k)f_n)}^{\ast }((w_n^k-1\otimes w_{n-1}^k)f_n)\hfill \\ \hfill & =f_n({(w_n^k)}^{\ast }{w}_n^k-{(w_n^k)}^{\ast }(1\otimes w_{n-1}^k)-{(1\otimes w_{n-1}^k)}^{\ast }{w}_n^k+1\otimes {(w_{n-1}^k)}^{\ast }{w}_{n-1}^k)f_n\hfill \\ \hfill & =f_n(2-{(w_n^k)}^{\ast }(1\otimes w_{n-1}^k)-{(1\otimes w_{n-1}^k)}^{\ast }{w}_n^k)f_n.\hfill \end{array}$$

Using the decomposition of $w_n^k$ from (20), we compute,

$$\begin{array}{c}\hfill {(w_n^k)}^{\ast }(1\otimes w_{n-1}^k)={\left(\frac{\varphi (n)}{\varphi (n+1)}\right)}^{\frac{1}{2}}·Q_k^{\otimes n}+\sum _{l=0}^{n-1}{1}^{\otimes l}\otimes (1-Q_k)\otimes Q_k^{\otimes n-1-l},\end{array}$$

which can be written as a matrix

$$\begin{array}{cc}\hfill {(w_n^k)}^{\ast }(1\otimes w_{n-1}^k)& =\left[\begin{array}{ccccc}{\left(\frac{\varphi (n)}{\varphi (n+1)}\right)}^{\frac{1}{2}}& 0& 0& \cdots & 0\\ 0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & 0\\ 0& 0& 0& \cdots & 0\end{array}\right],\hfill \end{array}$$

acting on $\text{Im}(Q_k^{\otimes n})\oplus {⨁}_{l=0}^{n-1}\text{Im}(1^{\otimes l}\otimes (1-Q_k)\otimes Q_k^{\otimes n-1-l})$.

Applying the same reasoning to ${(1\otimes w_{n-1}^k)}^{\ast }{w}_n^k$ yields

$$\begin{array}{c}\hfill \begin{array}{c}{f}_n(2-{(w_n^k)}^{\ast }(1\otimes w_{n-1}^k)-{(1\otimes w_{n-1}^k)}^{\ast }{w}_n^k)f_n=\\ f_n\left[\begin{array}{ccccc}2-2\sqrt{\frac{\varphi (n)}{\varphi (n+1)}}& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ 0& 0& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & 0\\ 0& 0& 0& \cdots & 0\end{array}\right]f_n,\end{array}\end{array}$$

acting on $\text{Im}(Q_k^{\otimes n})\oplus {⨁}_{l=0}^{n-1}\text{Im}(1^{\otimes l}\otimes (1-Q_k)\otimes Q_k^{\otimes n-1-l})$.

Therefore, we obtain

$$\begin{array}{cc}\hfill \Vert f_n(2-{(w_n^k)}^{\ast }(1\otimes w_{n-1}^k)-{(1\otimes w_{n-1}^k)}^{\ast }{w}_n^k)f_n{\Vert }^2& =2\left(1,-,\sqrt{\frac{\varphi (n)}{\varphi (n+1)}}\right)\hfill \\ \hfill & =2(1-{(1-{[n+1]}_q^{-2})}^{\frac{1}{2}}),\hfill \end{array}$$

which converges to zero as desired. $\square$

In particular, we obtain the following explicit KK-equivalence result.

Theorem 6.11

Let $P_1,P_2$ be two Temperley–Lieb polynomials, and let ${\mathbb{T}}_{⟨P_1,P_2⟩}$ be the associated free-product Toeplitz algebra. Denote by $i:\mathbb{C}\to {\mathbb{T}}_{⟨P_1,P_2⟩}$ the natural inclusion. The interior Kasparov product $[i]{\otimes }_{\mathbb{T}}[{\psi }_{+},{\psi }_{-}]$ agrees with the unit ${\mathbf{1}}_{\mathbb{C}}\in KK(\mathbb{C},\mathbb{C})$. In particular, [i] and $[{\psi }_{+},{\psi }_{-}]$ implement the KK-equivalence between $\mathbb{C}$ and $\mathbb{T}$.

Proof

The interior Kasparov product $[i]{\otimes }_{\mathbb{T}}[{\psi }_{+},{\psi }_{-}]\in KK(\mathbb{C},\mathbb{C})$ is represented by the pair $({\psi }_{+}\circ i,{\psi }_{-}\circ i)$, where ${\psi }_{\pm }\circ i:\mathbb{C}\to L({\mathcal{F}}^{\oplus 3})$ are $\ast$-homomorphisms.

In particular, ${\psi }_{+}\circ i$ is unital and

$$\begin{array}{c}\hfill {\psi }_{-}\circ i(1)=W_R{W}_R^{\ast }:{\mathcal{F}}^{\oplus 3}\to {\mathcal{F}}^{\oplus 3}\end{array}$$

is the orthogonal projection with range ${\mathcal{F}}_{+}\oplus {\mathcal{F}}^{\oplus 2}$.

Therefore, $({\psi }_{+}\circ i-{\psi }_{-}\circ i)(1)$ is the rank one orthogonal projection onto $\mathbb{C}\oplus \{0\}\subset {\mathcal{F}}^{\oplus 3}$.

Since we have already proven an abstract KK-equivalence between $\mathbb{C}$ and $\mathbb{T}$ in Theorem 5.13, and [i] maps the generators to generators, the claim follows. $\square$

Recall that the defining extension of Cuntz–Pimsner algebras of subproduct systems of finite-dimensional Hilbert spaces (1) induces a six-term exact sequence in K-theory: Observe that the algebra of compact operators is Morita equivalent to the complex numbers via the Fock space $\mathcal{F}$ of the subproduct system. We shall denote the KK-class of the Fock space and its dual by $[\mathcal{F}]\in KK(\mathbb{K}(\mathcal{F}),\mathbb{C})$ and $[{\mathcal{F}}^{\ast }]\in KK(\mathbb{C},\mathbb{K}(\mathcal{F}))$, respectively.

Proposition 6.12

Let $P_1$ and $P_2$ be Temperley–Lieb polynomials. Denote by $\mathcal{H}={\mathcal{H}}_{P_1}\star {\mathcal{H}}_{P_2}$ the free product of their subproduct systems. The following identity

$$\begin{array}{c}\hfill [j]{\otimes }_{\mathbb{T}}[{\psi }_{+},{\psi }_{-}]=[{\mathcal{F}}_{\mathcal{H}}]{\otimes }_{\mathbb{C}}\left({\mathbf{1}}_{\mathbb{C}},-,[H_1],+,[H_2^{\perp }]\right)\end{array}$$

holds in $KK(\mathbb{K},\mathbb{C})$.

Proof

Since $H_2^{\perp }$ is 2-dimensional, it is sufficient to show that

$$\begin{array}{c}\hfill [j]{\otimes }_{\mathbb{T}}[{\psi }_{+},{\psi }_{-}]=3·[{\mathcal{F}}_{\mathcal{H}}]-[{\mathcal{F}}_{\mathcal{H}}]{\otimes }_{\mathbb{C}}[H_1].\end{array}$$

The proof is a simple adaptation of that of [5, Proposition 7.1]. $\square$

The above proposition, combined with KK-equivalence proven in Theorem 6.11, yields

24

where $q\circ i:\mathbb{C}\to {\mathbb{O}}_{⟨P_1,P_2⟩}$.

Corollary 6.13

Let $P_1$ and $P_2$ be Temperley–Lieb polynomials in $d_1$ and $d_2$ variables, respectively. Then

$$K_0({\mathbb{O}}_{⟨P_1,P_2⟩})\cong \mathbb{Z}/(d_1+d_2-1)\mathbb{Z},K_1({\mathbb{O}}_{⟨P_1,P_2⟩})\cong \{0\}.$$

Note that the above K-groups are unchanged when one swaps $d_1$ with $d_2$.

By induction, we can extend this result to the case of finitely many Temperley–Lieb polynomials $P_1,\cdots ,P_r$, where the Fock space ${\mathcal{F}}_{\mathcal{H}}$ is the free product of ${\mathcal{F}}_{{\mathcal{H}}_i},i=1,\cdots ,r$, ${\mathcal{H}}_i$ denotes the subproduct system associated with Temperley–Lieb polynomial $P_i$, and the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}}$ is $\ast$-isomorphic to the reduced free product of the algebras ${\mathbb{T}}_{{\mathcal{H}}_i},i=1,\cdots ,r$.

Corollary 6.14

For $i=1,\cdots ,r$, let $P_i$ be a Temperley–Lieb polynomial in $d_i$ variables. Then

$$K_0({\mathbb{O}}_{⟨P_1,\cdots ,P_r⟩})\cong \mathbb{Z}/(\sum _{i=1}^r{d}_i-1)\mathbb{Z},K_1({\mathbb{O}}_{⟨P_1,\cdots ,P_r⟩⟩})\cong \{0\}.$$

Subproduct Systems from Quantum Group Corepresentations

Our interest in the representation theory of SU(2) and of its quantum counterpart, Woronowicz’s $S{U}_q(2)$, stems from their importance in various fields within mathematical physics, where they play a crucial role both in the study of symmetries and in quantum mechanics.

In [5], the authors gave a recipe for constructing a subproduct system of finite-dimensional Hilbert spaces starting from a finite-dimensional representation of the compact group SU(2) on a Hilbert space V. In their construction, the main ingredient was the so-called determinant of the representation, a subspace of the vector space $V\otimes V$. We will provide here an alternative and more compact definition for that notion. The authors would like to thank Marcel de Jeu for pointing this out to us.

Definition 7.1

Let $(\rho ,V)$ be a finite-dimensional unitary representation of the group SU(2). Define the determinant of the representation $(\rho ,V)$ as the isotypical component of the trivial representation in $(\rho \otimes \rho ,V\otimes V)$.

This definition can be dualised to the case of a corepresentation of the Hopf $C^{\ast }$-algebra $S{U}_q(2)$, and more generally, to the setting of a corepresentation of a rank-two compact quantum group. We assume the reader to be familiar with the relevant notions from the theory of quantum group corepresentations [25], in particular with the tensor product of two corepresentations.

Definition 7.2

Let $\rho :V\to V\otimes C(S{U}_q(2))$ be a right corepresentation of the quantum group $S{U}_q(2)$ on V. We define the determinant of $\rho$ as the isotypical component of the trivial corepresentation in the diagonal corepresentation $\rho \otimes \rho$ on the tensor product $H\otimes H$:

$$\text{det}(\rho ,V)=\{\xi \in V\otimes V\mid (\rho \otimes \rho )(\xi )=\xi \otimes 1\}.$$

Note that since the determinant is a subspace of $H\otimes H$, taking its orthogonal complement gives a quadratic subproduct system of Hilbert spaces.

Example 7.3

Recall that the fundamental corepresentation ${\rho }_1:{\mathbb{C}}^2\to {\mathbb{C}}^2\otimes C(S{U}_q(2))$, i.e. the irreducible corepresentation of $S{U}_q(2)$ with highest weight 1, has matrix coefficients

$$\left(\begin{array}{cc}a& -q{c}^{\ast }\\ c& a^{\ast }\end{array}\right).$$

Let us consider the standard basis of ${\mathbb{C}}^2$. It is easy to check that the determinant is spanned by the Temperley–Lieb vector

$$\begin{array}{c}\hfill q^{-1/2}{e}_1\otimes e_2-q^{1/2}{e}_2\otimes e_1.\end{array}$$ 25

This follows from the commutation relations of $S{U}_q(2)$, in particular

$$\begin{array}{c}\hfill a{c}^{\ast }=q{c}^{\ast }a,a^{\ast }a+c^{\ast }c=1=a{a}^{\ast }+q^{2}c{c}^{\ast },\end{array}$$

together with the fact that c is normal.

Note that $det({\rho }_1,{\mathbb{C}}^2)$ is nothing but the q-antisymmetric subspace of ${\mathbb{C}}^2\otimes {\mathbb{C}}^2$ defined in [28] using the braiding ${\sigma }_q$ given by

$$\left\{\begin{array}{cc}{\sigma }_q(e_i\otimes e_i)=e_i\otimes e_i,\hfill & i=1,2\hfill \\ {\sigma }_q(e_1\otimes e_2)=q{e}_2\otimes e_1,\hfill & \\ {\sigma }_q(e_2\otimes e_1)=q{e}_1\otimes e_2+(1-q^2)e_2\otimes e_1.\hfill \end{array}\right)$$

Its orthogonal complement is the so-called q-symmetric tensor product.

It is a well-known fact that the group SU(2) and its quantum analogue $S{U}_q(2)$ have the same representation category, and hence the same fusion rules.

Theorem 7.4

[34, Theorem 5.11] Let $V_n,V_m$ be the irreducible corepresentation of $S{U}_q(2)$ with highest weights n and m, respectively. Then the tensor product of corepresentations $V_n\otimes V_m$ decomposes as

$$\begin{array}{c}\hfill V_n\otimes V_m\cong V_{|n-m|}\oplus V_{|n-m|+1}\oplus \cdots \oplus V_{n+m}.\end{array}$$

To characterize the determinant of $V_n$, we apply the Clebsch–Gordan formula [19, Equation (54)], obtaining

$$\begin{array}{c}\hfill \text{det}({\rho }_n,{\mathbb{C}}^{n+1})={\text{span}}_{\mathbb{C}}\left\{\sum _{i=1}^{n+1},{(-1)}^i,·,{\left(\frac{q^{-n+i}}{{[n+1]}_q}\right)}^{1/2},·,e_i,\otimes ,e_{n+2-i},,\right\}\end{array}$$ 26

where $\{e_1,e_2,\cdots ,e_{n+1}\}$ is an orthonormal basis of $V_n\cong {\mathbb{C}}^{n+1}$, see also [13]. The vector above is Temperley–Lieb [14, Lemma 1.4], with corresponding Temperley–Lieb polynomial.

$$\begin{array}{c}\hfill P(X_1,\cdots ,X_n)=\sum _{i=1}^{n+1}{(-1)}^i·{\left(\frac{q^{-n+i}}{{[n+1]}_q}\right)}^{1/2}·X_i{X}_{n+2-i}.\end{array}$$ 27

Therefore, the $S{U}_q(2)$-subproduct system is a Temperley–Lieb subproduct system.

Lemma 7.5

Let $\rho$ be a finite-dimensional corepresentation of $S{U}_q(2)$, then the determinant has dimension equal to the sum of the squares of the multiplicities of its irreducible components.

Proof

Let $\mathbf{1}$ denote the trivial corepresentation. By Definition 7.2. $det(\rho )$ is the isotypical component of $\mathbf{1}$ in $(\rho \otimes \rho ,H\otimes H)$, and it is thus determined by the intertwiner space $\text{Hom}(\rho \otimes \rho ,\mathbf{1})$.

Let $\rho$ be a finite-dimensional reducible corepresentation of $S{U}_q(2)$. Then $\rho$ decomposes into the direct sum of irreducible corepresentations ${\rho }_n$ of highest weight n, with multiplicity $k_n$, i.e., $\rho ={⨁}_{n=0}^{\infty }{\rho }_n^{\oplus k_n}$ with finitely many $k_n$’s non-zero. We compute

$$\text{Hom}(\rho \otimes \rho ,\mathbf{1})\cong \text{Hom}(\rho ,{\rho }^{\ast })\cong \text{Hom}\left(\underset{n}{⨁},{\rho }_n^{\oplus k_n},,,\underset{n}{⨁},{({\rho }_n^{\oplus k_n})}^{\ast }\right).$$

Given that ${\rho }_n$ is not equivalent to ${\rho }_m$ for $n\ne m$, and irreducible corepresentations of $S{U}_q(2)$ are self-dual, we obtain

$$\text{Hom}(\rho \otimes \rho ,\mathbf{1})\cong \underset{n}{⨁}\text{Hom}\left({\rho }_n^{\oplus k_n},,,{({\rho }_n^{\oplus k_n})}^{\ast }\right)\cong \underset{n}{⨁}\text{Hom}({\rho }_n^{\oplus k_n}\otimes {\rho }_n^{\oplus k_n},\mathbf{1}).$$

We deduce that

$$\begin{array}{cc}\hfill dim(\text{Hom}(\rho \otimes \rho ,\mathbf{1}))& =\sum _{n=0}^{\infty }dim(\text{Hom}({\rho }_n^{\oplus k_n}\otimes {\rho }_n^{\oplus k_n},\mathbf{1}))\hfill \\ \hfill & =\sum _{n=0}^{\infty }dim(\text{Hom}({({\rho }_n\otimes {\rho }_n)}^{\oplus k_n^2},\mathbf{1}))=\sum _{n=0}^{\infty }{k}_n^2,\hfill \end{array}$$

$\square$

The Subproduct System of a Multiplicity-Free Corepresentation

Theorem 7.6

Let $(\rho ,H)$ be a finite-dimensional multiplicity-free corepresentation of $S{U}_q(2)$. The $S{U}_q(2)$-subproduct system of $\rho$ is isomorphic to the free product of the $S{U}_q(2)$-subproduct systems of its irreducible components. Correspondingly, the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}}$ is the reduced free product of the Toeplitz algebras of the subproduct systems of its irreducible components.

Proof

To establish the result, it suffices to show that the determinant of a multiplicity-free unitary representation is spanned by Temperley–Lieb vectors.

By definition, $det(\rho )$ is the isotypical component of the trivial corepresentation within $(\rho \otimes \rho ,H\otimes H)$ and is thus determined by the intertwiner space $\text{Hom}(\rho \otimes \rho ,\mathbf{1})$. Since $\rho$ is multiplicity-free, a similar argument to that in Lemma 7.5 gives

$$\begin{array}{c}\hfill \text{Hom}(\rho \otimes \rho ,\mathbf{1})\cong {\oplus }_i\text{Hom}({\rho }_{n_i}\otimes {\rho }_{n_i},\mathbf{1}).\end{array}$$

Consequently, the determinant of the representation $\rho$ decomposes as the direct sum of the determinants of its irreducible components.

Let us decompose $\rho$ into its irreducible components, i.e. $\rho ={\rho }_{n_1}\oplus \cdots \oplus {\rho }_{n_r}$, where ${\rho }_{n_i}$ denotes the irreducible corepresentation with highest weight $n_i$ and ${\rho }_{n_k}\ncong {\rho }_{n_l}$ for $k\ne l$. Denote by ${\mathcal{H}}^i$ the $S{U}_q(2)$ subproduct system associated with ${\rho }_{n_i}$ and by $\mathcal{H}$ the $S{U}_q(2)$ subproduct system associated with $\rho$. For each irreducible component ${\rho }_{n_i}$, the determinant is one-dimensional and spanned by the vector in 26. Therefore, the determinant of $\rho$ is spanned by a union of independent Temperley–Lieb vectors. $\square$

Since the Toeplitz algebra associated with an irreducible representation of $S{U}_q(2)$ is nuclear [14, Corollary 3.3], combining Theorem 7.6 with Theorem 5.13, we deduce the following:

Corollary 7.7

Let $(\rho ,H)$ be a finite-dimensional multiplicity-free corepresentation of $S{U}_q(2)$, and $\mathcal{H}$ be the associated $S{U}_q(2)$-subproduct system. Then the Toeplitz algebra ${\mathbb{T}}_{\mathcal{H}}$ is nuclear.

Remark 7.8

By analogy with the Cuntz–Pimsner case [18], the above nuclearity result should not come as a surprise to the reader. Presently, we are not aware of examples of Toeplitz algebras of subproduct systems of Hilbert spaces that do not satisfy nuclearity. However, subproduct systems of correspondences over arbitrary C$_{}^{\ast }$-algebras may be broad enough to incorporate non-nuclear examples.

From the fact that the subproduct system of an irreducible $S{U}_q(2)$-representation is Temperley–Lieb, we may view the exact sequence (24) as a noncommutative Gysin sequence [5]. To this end, we define the Euler class of the representation to be

$$\begin{array}{c}\hfill \chi (\rho )=\chi \left({\oplus }_{i=1}^r,{\rho }_{n_i}\right):={\mathbf{1}}_{\mathbb{C}}-[H_1({\oplus }_{i=1}^r{\rho }_{n_i})]+[det({\oplus }_{i=1}^r{\rho }_{n_i})]\in KK(\mathbb{C},\mathbb{C}).\end{array}$$

Theorem 7.9

We have an exact sequence of groups: Therefore, we have

$$\begin{array}{cc}& K_1(\mathbb{O})\cong ker\left({\mathbf{1}}_{\mathbb{C}},-,[E_1({\oplus }_{i=1}^r{\rho }_i)],+,[det({\oplus }_{i=1}^r{\rho }_i)],)\right),\hfill \\ \hfill & K_0(\mathbb{O})\cong \text{coker}\left({\mathbf{1}}_{\mathbb{C}},-,[E_1({\oplus }_{i=1}^r{\rho }_i)],+,[det({\oplus }_{i=1}^r{\rho }_i)],)\right).\hfill \end{array}$$

More precisely, for $\rho \cong {\oplus }_{i=1}^r{\rho }_{n_i}$, the K-theory groups of its Cuntz–Pimsner algebra are

$$\begin{array}{c}\hfill K_0(\mathbb{O})\cong \mathbb{Z}/(\sum _{i=1}^r{n}_i-1)·\mathbb{Z}{K}_1(\mathbb{O})\cong \left\{\begin{array}{c}\mathbb{Z}r=1,n_1=1\hfill \\ 0\text{otherwise}.\hfill \end{array}\right)\end{array}$$

Remark 7.10

The above theorem extends [5, Corollary 7.3] beyond the irreducible case.

Dealing with Multiplicities

Let $\rho$ be an isotypical corepresentation of $S{U}_q(2)$ with highest weight n and multiplicity t, i.e. $\rho ={\rho }_n^{\oplus t}$, then by Lemma 7.5, we have $dim(det(\rho ))=t^2$. Indeed, an explicit basis for $det(\rho )$ is the following:

$$\begin{array}{c}\hfill \text{det}(\rho )=\left\{\sum _{i=1}^{n+1},{(-1)}^i,q^{i/2},e_i^k,\otimes ,e_{n+1-i}^l,:,k,,,l,=,1,,,2,,,\cdots ,,,t\right\},\end{array}$$

where we denote

$$\begin{array}{c}\hfill e_i^k=0\oplus \cdots 0\oplus \underset{k\text{th}}{\underbrace{e_i}}\oplus 0\oplus \cdots \oplus 0,\end{array}$$

and the common divisor of the coefficient ${[n+1]}_q^{-1/2}$ is omitted.

Remark 7.11

The subproduct system of an isotypical representation is a quadratic subproduct system with few relations. Indeed, let $dim(\rho )=m$ then we have mt generators and $t^2$ relations, and it is easy to see that $t^2\le {(mt)}^2/4$ precisely when $m\ge 2$.

Let $n\in \mathbb{N}$. By ${\rho }_n$ we mean the irreducible corepresentation of $S{U}_q(2)$ of highest weight n. For simplicity, we denote the associated $S{U}_q(2)$-subproduct system $\mathcal{H}:={\{H_m\}}_{m\in {\mathbb{N}}_0}$, where $H_1$ is the representation space. Moreover, we denote the $S{U}_q(2)$-subproduct system associated with the representation $\rho ={\rho }_n^{\oplus t}$ by $\{H_m^t:m\in \mathbb{N}\}$. For any $1\le k\le t$, we define ${\sigma }_k^1:H_1\to H_1^t$ as the $S{U}_q(2)$-equivariant linear maps given on the basis vectors by ${\sigma }_k^1(e_i)=e_i^k$.

By definition, $H_1^t\cong H_1^{\oplus t}$ through the $S{U}_q(2)$-equivariant isomorphism given by

$$\begin{array}{c}\hfill H_1^t\cong {\sigma }_1^1(H_1)\oplus {\sigma }_2^1(H_1)\oplus \cdots \oplus {\sigma }_t^1(H_1).\end{array}$$

The vector space $H_m^t$ can be described in a similar way:

Proposition 7.12

Let ${\rho }_n$ denote the irreducible corepresentation of highest weight n and $\mathcal{H}$ be the corresponding $S{U}_q(2)$-subproduct system. Let ${\mathcal{H}}^t$ be the subproduct system of the corepresentation ${\rho }_n^{\oplus t}$. There is a unitary isomorphism:

$$\begin{array}{c}\hfill H_m^{\oplus t^m}\simeq H_m^t\end{array}$$ 29

Proof

We will show that the isomorphism is implemented by the map

$$\underset{k_1,k_2,\cdots ,k_m=1}{\overset{t}{⨁}}{\sigma }_1^1\otimes {\sigma }_2^1\otimes \cdots \otimes {\sigma }_{k_m}^1:H_m^{\oplus t^m}\to H_m^t.$$

We prove this by induction. The statement is true for $n=1$. For $n=2$, recall the definition of $H_2^t$ as the orthogonal complement of the determinant in $H_1^t\otimes H_1^t$. Observe that

$$\begin{array}{c}\hfill H_1^t\otimes H_1^t\cong \underset{k_1,k_2=1}{\overset{t}{⨁}}{\sigma }_1^1(H_1)\otimes {\sigma }_2^1(H_1).\end{array}$$

Moreover, we have that

$$\begin{array}{c}\hfill D:=det({\rho }_n^{\oplus t})\cong \underset{k_1,k_2=1}{\overset{t}{⨁}}{\sigma }_1^1\otimes {\sigma }_2^1(det({\rho }_n)).\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill H_2^t& =D^{\perp }\cong \underset{k_1,k_2=1}{\overset{t}{⨁}}{\sigma }_1^1(H_1)\otimes {\sigma }_2^1(H_1)\ominus \underset{k_1,k_2=1}{\overset{t}{⨁}}{\sigma }_1^1\otimes {\sigma }_2^1({\text{det}}_q({\rho }_n))\hfill \\ \hfill & \cong \underset{k_1,k_2=1}{\overset{t}{⨁}}{\sigma }_1^1\otimes {\sigma }_2^1(H_1\otimes H_1\ominus det({\rho }_n))\cong \underset{k_1,k_2=1}{\overset{t}{⨁}}{\sigma }_1^1\otimes {\sigma }_2^1(H_2),\hfill \end{array}$$

which proves the claim for $m=2$.

Using the recursive formula in Remark 3.15, we obtain

$$\begin{array}{cc}& H_{m+1}^t\cong H_1^t\otimes H_m^t\cap H_m^t\otimes H_1^t\hfill \\ \hfill & \cong \underset{k,k_1,\cdots ,k_m=1}{\overset{t}{⨁}}{\sigma }_k^1\otimes {\sigma }_1^1\otimes \cdots \otimes {\sigma }_{k_m}^1(H_1\otimes H_m\cap H_m\otimes H_1)\hfill \\ \hfill & \cong \underset{k,k_1,\cdots ,k_m=1}{\overset{t}{⨁}}{\sigma }_k^1\otimes {\sigma }_1^1\otimes \cdots \otimes {\sigma }_{k_m}^1(H_{m+1}).\hfill \end{array}$$

$\square$

Corollary 7.13

Let ${\rho }_n$ be the irreducible $S{U}_q(2)$ corepresentation with highest weight n, and let $h_n(z)$ be the Hilbert series of the associated $S{U}_q(2)$ subproduct system. The Hilbert series of the subproduct system of the isotypical corepresentation ${\rho }_n^{\oplus t}$ satisfies

$$\begin{array}{c}\hfill h_{\mathit{tn}}(z)={(1-t(n+1)z+t^2{z}^2)}^{-1}=h_n(tz)\end{array}$$ 30

Proof

The proof follows from the corresponding claim for dimension sequences: let $d^{(n)}$ be the dimension sequence of the subproduct system of the irreducible corepresentations ${\rho }_n$. Then the subproduct system of the isotypical corepresentation ${\rho }_n^{\oplus t}$ is given by $d_m^{(n,t)}:=d_m^{(n)}{t}^m.$ Our claim then follows from the definition of Hilbert series. $\square$

Combining this result with Remark 7.11, we obtain the following:

Corollary 7.14

Let ${\rho }_n$ be the irreducible $S{U}_q(2)$-corepresentation of highest weight n. The subproduct system of the corepresentation ${\rho }_n^{\oplus t}$ is a generic quadratic subproduct system in $t(n+1)$ generators and $t^2$ relations.

Example 7.15

Let ${\rho }_1$ be the fundamental corepresentation on $V_1\simeq {\mathbb{C}}^2$ with orthonormal basis $\{e_1,e_2\}$. Then the determinant is given by (25).

Consider the isotypical corepresentation ${\rho }_1^{\oplus 2}$ on $H_1^t$ with orthonormal basis $\{e_1^1,e_2^1,e_1^2,e_2^2\}$. Then we have $det({\rho }_1^{\oplus 2})\cong det{({\rho }_1)}^{\oplus 4}$, which is spanned by

$$\begin{array}{c}\hfill \{q^{1/2}·e_1^k\otimes e_2^l-q^{-1/2}·e_2^k\otimes e_1^l:k,l=1,2\}.\end{array}$$

The space $H_2^t\cong H_2^{\oplus 4}$ is spanned by

$$\begin{array}{c}\hfill \{e_1^k\otimes e_1^l,e_2^k\otimes e_2^l,q^{-1/2}·e_1^k\otimes e_2^l+q^{1/2}·e_2^k\otimes e_1^l:k,l=1,2\}.\end{array}$$

As discussed earlier, this construction gives a generic quadratic subproduct system with few relations, with Hilbert series

$$\begin{array}{c}\hfill h(z)={(1-4z+4z^2)}^{-1}.\end{array}$$

Outlook

It is natural to wonder what operation in the algebraic world of associative algebras corresponds to the change of variable in the Hilbert series described in (30), and to consider what the consequences of this operation are at the level of the Toeplitz algebras.

Finally, it seems that K-theory computations only read the Hilbert series of a quadratic algebra and that Cuntz–Pimsner algebras of non-isomorphic subproduct systems with the same Hilbert series are KK-equivalent. We postpone the discussion of these and other related questions to future work.

Funding

This work is partially funded by the Netherlands Organisation of Scientific Research (NWO) under grants 016.Veni.192.237, and VI.Vidi.233.231. Part of the EU Staff Exchange project 101086394 “Operator Algebras That One Can See”.

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Declarations

Competing interest

The authors have no competing interests to declare that are relevant to the content of this article.