Operator algebras associated with unitary commutation relations

TL;DR

This paper introduces a class of nonselfadjoint operator algebras generated by unitary commutation relations, generalizing analytic Toeplitz algebras of rank 2 graphs, and classifies them via an associated unitary matrix.

Contribution

It defines and classifies new operator algebras based on unitary commutation relations, extending the theory of analytic Toeplitz algebras of rank 2 graphs.

Findings

Algebras are classified up to isometric isomorphism by the unitary matrix u.

Generalization of analytic Toeplitz algebras of rank 2 graphs.

Provides a framework for understanding operator algebras with unitary commutation relations.

Abstract

We define nonselfadjoint operator algebras with generators $L_{e_1},..., L_{e_n}, L_{f_1},...,L_{f_m}$ subject to the unitary commutation relations of the form \[ L_{e_i}L_{f_j} = \sum_{k,l} u_{i,j,k,l} L_{f_l}L_{e_k}\] where $u= (u_{i,j,k,l})$ is an $nm \times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.

Equations

$$L_{e_i}{L}_{f_j}=\sum _{k,l}{u}_{i,j,k,l}{L}_{f_l}{L}_{e_k}$$

$$e_i\otimes f_j=\sum _{k,l}{u}_{(i,j),(k,l)}{f}_l\otimes e_k.$$

$$f_l\otimes e_k=\sum _{i,j}{\bar{u}}_{(i,j),(k,l)}{e}_i\otimes f_j.$$

$$\sum _{k,l}X(k,l)=\sum _{k,l}{E}^{\otimes k}\otimes F^{\otimes l},$$

$$L_e{e}_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{j_l}=e\otimes e_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{j_l}$$

$$L_f{e}_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{j_l}=f\otimes e_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{j_l}$$

$${\Sigma }_p(A)=\sum _{k\le p}(1-\frac{k}{p}){\Phi }_k(A)$$

$${\Phi }_k(a)=\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-ikt}{\gamma }_t(a)dt$$

$$R_e{e}_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{j_l}=e_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{j_l}\otimes e$$

$$R_f{e}_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{i_l}=e_{i_1}\otimes e_{i_2}\otimes \cdots \otimes e_{i_k}\otimes f_{j_1}\otimes f_{j_2}\otimes \cdots \otimes f_{i_l}\otimes f.$$

$$W:\mathcal{F}(n,m,u)\to \mathcal{F}(n,m,u^{\ast })$$

$$V_u=\{(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m:z_i{w}_j=\sum _{k,l}{u}_{(i,j),(k,l)}{z}_k{w}_l\}$$

$${\Omega }_u=V_u\cap ({\overline{\mathbb{B}}}_n\times {\overline{\mathbb{B}}}_m)$$

$$w_{\alpha }=\sum _{i,j}\sum _{k,l}\overline{\alpha (e_i{f}_j)}{e}_i{f}_j\in \mathcal{F}(X).$$

$$\Vert w_{\alpha }{\Vert }^2=\sum _{i,j,k,l}|\alpha (e_i{f}_j){|}^2=(1-\Vert z{\Vert }^2{)}^{-1}(1-\Vert w{\Vert }^2{)}^{-1}<\infty$$

$$⟨x,w_{\alpha }⟩=\alpha (x).$$

$$\alpha (x)=⟨L_x{\nu }_{\alpha },{\nu }_{\alpha }⟩$$

$$\sum _{k,l}{u}_{(i,j),(k,l)}{z}_k{w}_l=\sum _k(\sum _l{u}_{(i,j),(k,l)}{w}_l)z_k=\sum _k(u_{(i,j)}w{)}_k{z}_k=⟨u_{(i,j)}w,\bar{z}⟩.$$

$$V_u=\{(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m:⟨C_{(i,j)}w,\bar{z}⟩=0,\text{mbox}foralli,j\}.$$

$${\Omega }_u^0:=\{(z,w)\in {\overline{\mathbb{B}}}_n\times {\overline{\mathbb{B}}}_m:C_{(i,j)}w=0,C_{(i,j)}^{t}z=0\text{mbox}foralli,j\}.$$

$$\sum _l{u}_{(i,j),(k,l)}{w}_l={\delta }_{i,k}{w}_j$$

$$\sum _k{u}_{(i,j),(k,l)}{z}_k={\delta }_{j,l}{z}_i$$

$$\sum _k{u}_{(k,l),(i,j)}\overline{z_k}={\delta }_{j,l}\overline{z_i}.$$

$$span\{{\tilde{z}}_{(j)},{\tilde{w}}^{(i)}:1\le i\le n,1\le j\le m\}\subseteq Ker(u-I).$$

$$\rho(T)=\left(\begin{array}[]{cc}\rho_{1,1}(T)&\rho_{1,2}(T)\\ 0&\rho_{2,1}(T)\end{array}\right)$$

$${\rho }_{1,2}(TS)={\rho }_{1,1}(T){\rho }_{1,2}(S)+{\rho }_{1,2}(T){\rho }_{2,2}(S)$$

$$z_i{\mu }_j+{\lambda }_i{w}_j=\sum _{k,l}{u}_{(i,j),(k,l)}(w_l{\lambda }_k+{\mu }_l{z}_k).$$

$$⟨u_{(i,j)}w,\bar{\lambda }⟩+⟨u_{(i,j)}\mu ,\bar{z}⟩=z_i{\mu }_j+{\lambda }_i{w}_j=⟨E_{i,j}w,\bar{\lambda }⟩+⟨E_{i,j}\mu ,\bar{z}⟩.$$

$$⟨C_{(i,j)}w,\bar{\lambda }⟩+⟨\mu ,\overline{C_{(i,j)}^{t}z}⟩=0.$$

Full text

$L_{e_{1}},\dots,L_{e_{n}},L_{f_{1}},\dots,L_{f_{m}}$ subject to the unitary commutation relations of the form

[TABLE]

where $u=(u_{i,j,k,l})$ is an $nm\times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.

1 Introduction

The unilateral shift on complex separable Hilbert space generates two fundamental operator algebras, namely the norm closed (unital) algebra and the weak operator topology closed algebra. The former is naturally isomorphic to the disc algebra of holomorphic functions on the unit disc, continuous to the boundary, while the latter is isomorphic to $H^{\infty}$. The freely noncommuting multivariable generalisations of these algebras arise from the freely noncommuting shifts $L_{e_{1}},\dots,L_{e_{n}}$ given by the left creation operators on the Fock space $\mathcal{F}_{n}=\sum_{k=0}^{\infty}\oplus({\mathbb{C}}^{n})^{\otimes k}.$ Here the generated operator algebras, denoted $\mathcal{A}_{n}$ and $\mathcal{L}_{n}$ for the norm and weak topologies, are known as the noncommutative disc algebra and the freesemigroup algebra. They have been studied extensively with respect to operator algebra structure, representation theory and the multivariable operator theory of row contractions. See for example [2], [9].

Higher rank generalisations of these algebras arise when one considers several families of freely noncommuting generators between which there are commutation relations. In the present paper we consider a very general form of such relations, namely

[TABLE]

where $L_{e_{1}},\dots,L_{e_{n}}$ and $L_{f_{1}},\dots,L_{f_{m}}$ are freely noncommuting and $u=(u_{i,j,k,l})$ is an $nm\times nm$ unitary matrix. The associated operator algebras are denoted $\mathcal{A}_{u}$ and $\mathcal{L}_{u}$ and we classify them up to various forms of isomorphism in terms of the unitary matrices $u$. Such unitary relations arose originally in the context of the general dilation theorem proven in Solel ([12], [13]) for two row contractions $[T_{1}\cdots T_{n}]$ and $[S_{1}\cdots S_{m}]$ satisfying the unitary commutation relations.

For $n=m=1$, we have $u=[\alpha]$ with $|\alpha|=1$ and $\mathcal{A}_{u}$ is the subalgebra of the rotation C*-algebra for the relations $uv=\alpha vu$. When $u$ is a permutation unitary matrix arising from a permutation $\theta$ in $S_{nm}$ then the relations are those associated with a single vertex rank 2 graph in the sense of Kumjian and Pask, and the algebras in this case have been considered in Kribs and Power [5] and Power [10]. In particular, in [10] it was shown that there are 9 operator algebras $\mathcal{A}_{\theta}$ arising from the 24 permutations in case $n=m=2$. In contrast, we see below in Section 6 that for general $2$ by $2$ unitaries $u$ there are uncountably many isomorphism classes of the unitary relation algebras $\mathcal{A}_{u}$ expressed in terms of a nine fold real parametrisation of isomorphism types.

The algebras $\mathcal{A}_{\theta}$ are easily defined; they are determined by the left regular representation of the semigroup $\mathbb{F}^{+}_{\theta}$ whose generators are $e_{1},\dots,e_{n},f_{1},\dots,f_{m}$ subject to the relations $e_{i}f_{j}=f_{l}e_{k}$ where $\theta(i,j)=(k,l)$. On the other hand the unitary relation algebras $\mathcal{A}_{u}$ are generated by creation operators on a ${\mathbb{Z}}_{+}^{2}$-graded Fock space $\sum_{k,l}\oplus({\mathbb{C}}^{n})^{\otimes k}\otimes({\mathbb{C}}^{m})^{\otimes l}$ with relations arising from the identification $u:{\mathbb{C}}^{n}\otimes{\mathbb{C}}^{m}\to{\mathbb{C}}^{m}\otimes{\mathbb{C}}^{n}$. In particular, $\mathcal{A}_{u}$ is a representation of the non-selfadjoint tensor algebra of a rank $2$ correspondence (or a product system over $\mathbb{N}^{2}$) in the sense of [13]. See also [3]

In the main results, summarised partly in Theorem 5.10, we see that if $\mathcal{A}_{u}$ and $\mathcal{A}_{v}$ are isomorphic then the two families of generators have matching cardinalities. Furthermore, if $n\neq m$ then the algebras are isomorphic if and only if the unitaries $u,v$ in $M_{nm}({\mathbb{C}})$ are unitary equivalent by a unitary $A\otimes B$ in $M_{n}({\mathbb{C}})\otimes M_{m}({\mathbb{C}})$. As in [10] we term this product unitary equivalence (with respect to the fixed tensor product decomposition). The case $n=m$ admits an extra possibility, in view of the possibility of generator exchanging isomorphisms, namely that $u,\tilde{v}$ are product unitary equivalent, where $\tilde{v}_{i,j,k,l}=\bar{v}_{l,k,j,i}$.

The theorem is proven as follows. After some preliminaries we identify, in Section 3, the character space $M(\mathcal{A}_{u})$ and the set of w*-continuous characters on $\mathcal{L}_{u}$. These are subsets of the closed unit ball product $\overline{{\mathbb{B}}}_{n}\times\overline{{\mathbb{B}}}_{m}$ which are associated with a variety $V_{u}$ in ${\mathbb{C}}^{n}\times{\mathbb{C}}^{m}$ determined by $u$. We then define the core $\Omega_{u}^{0}$, a closed subset of the realised character space $\Omega_{u}=M(\mathcal{A}_{u})$, and we identify this intrinsically (algebraically) in terms of representations of $\mathcal{A}_{u}$ into $T_{2}$, the algebra of upper triangular matrices in $M_{2}({\mathbb{C}})$. The importance of the core is that we are able to show that the interior is a minimal automorphism invariant subset on which automorphisms act transitively. This allows us to infer the existence of graded isomorphisms from general isomorphisms. To construct automorphisms we first review, in Section 4, Voiculescu’s construction of a unitary action of the Lie group $U(1,n)$ on the Cuntz algebra $\mathcal{O}_{n}$ and the operator algebras $\mathcal{A}_{n}$ and $\mathcal{L}_{n}$. This provides, in particular, unitary automorphisms $\Theta_{\alpha}$, for $\alpha\in{\mathbb{B}}_{n}$, which act transitively on the interior ball, ${\mathbb{B}}_{n}$, of the character space of $\mathcal{A}_{n}$. For these explicit unitary automorphisms of the $e_{i}$-generated copy of $\mathcal{A}_{n}$ in $\mathcal{A}_{u}$, we establish unitary commutation relations for the tuples $\Theta_{\alpha}(L_{e_{1}}),\dots,\Theta_{\alpha}(L_{e_{n}})$ and $L_{f_{1}},\dots,L_{f_{m}}$, when $(\alpha,0)$ is a point in the core. This enables us to define natural unitary automorphisms of $\mathcal{A}_{u}$ itself, and in Theorem 4.8 the relative interior of the core is identified as an automorphism invariant set in the Gelfand space $\Omega_{u}$. In Section 5 we determine the graded and bigraded isomorphisms in terms of product unitary equivalence. To do this we observe that such isomorphisms induce an origin preserving biholomorphic map between the cores $\Omega_{u}^{0}$ and $\Omega_{v}^{0}$ and that these maps, by a generalised Schwarz’s Lemma, are implemented by a product unitary. We then prove the main classification theorem.

In Section 6 we analyse in detail the case $n=m=2$ and consider the special case of permutation unitaries.

Finally, in Section 7 we show that the algebra $\mathcal{A}_{u}$ is contained in a tensor algebra $\mathcal{T}_{+}(X)$, associated with a correspondence $X$ as in [7]. Moreover, at least when $n\neq m$, every automorphism of $\mathcal{A}_{u}$ extends to an automorphism of $\mathcal{T}_{+}(X)$. The advantage of the tensor algebra is that its representation theory is known ([7]) while this is not the case yet for the algebra $\mathcal{A}_{u}$.

2 Preliminaries

Fix two finite dimensional Hilbert spaces $E=\mathbb{C}^{n}$ and $F=\mathbb{C}^{m}$ and a unitary $mn\times mn$ matrix $u$. The rows and columns of $u$ are indexed by $\{1,\ldots,n\}\times\{1,\ldots,m\}$ ($u=(u_{(i,j),(k,l)})$) and when we write $u$ as an $mn\times mn$ matrix we assume that $\{1,\ldots,n\}\times\{1,\ldots,m\}$ is ordered lexicographically (so that, for example, the second row is the row indexed by $(1,2)$). We also fix orthonormal bases $\{e_{i}\}$ and $\{f_{j}\}$ for $E$ and $F$ respectively and the matrix $u$ is used to identify $E\otimes F$ with $F\otimes E$ through the equation

[TABLE]

Equivalently, we write

[TABLE]

For every $k,l\in\mathbb{N}$, we write $X(k,l)$ for $E^{\otimes k}\otimes F^{\otimes l}$. Using succesive applications of (1), we can identify $X(k,l)$ with $E^{\otimes k_{1}}\otimes F^{\otimes l_{1}}\otimes E^{\otimes k_{2}}\otimes\cdots\otimes F^{\otimes l_{r}}$ whenever $k=\sum k_{i}$ and $l=\sum l_{j}$.

Let $\mathcal{F}(n,m,u)$ be the Fock space given by the Hilbert space direct sum

[TABLE]

and, for $e\in E$ and $f\in F$, write $L_{e}$ and $L_{f}$ for the “shift” operators

[TABLE]

and

[TABLE]

where, in the last equation, we use (1) to identify the resulting vector as a vector of $E^{\otimes k}\otimes F^{\otimes(l+1)}$.

The unital semigroup generated by $\{I,L_{e},L_{f}\;:\;e\in E,\;f\in F\}$ is denoted $\mathbb{F}^{+}_{u}$ and the algebra it generates denoted $\mathbb{C}[\mathbb{F}_{u}^{+}]$. The norm closure of $\mathbb{C}[\mathbb{F}_{u}^{+}]$ will be written $\mathcal{A}_{u}$ and its closure in the weak* operator topology will be written $\mathcal{L}_{u}$. In particular, the algebras $\mathcal{L}_{\theta}$ and $\mathcal{A}_{\theta}$ studied in [10] are the algebras $\mathcal{L}_{u}$ and $\mathcal{A}_{u}$ for $u$ which is a permutation matrix.

The results of Section 2 in [5] hold here too with minor changes. Every $A\in\mathcal{L}_{u}$ is the limit (in the strong operator topology) of its Cesaro sums

[TABLE]

where $\Phi_{k}(A)$ lies in $\mathcal{L}_{u}$ and is “supported” on $\sum_{l}\oplus E^{\otimes l}\otimes F^{\otimes(k-l)}$. In fact, let $Q_{k}$ be the projection of $\mathcal{F}(n,m,u)$ onto $\sum_{l}\oplus E^{\otimes l}\otimes F^{\otimes(k-l)}$, form the one-parameter unitary group $\{U_{t}\}$ defined by $U_{t}:=\sum_{k=0}^{\infty}e^{ikt}Q_{k}$ and set $\gamma_{t}=AdU_{t}$. Then $\{\gamma_{t}\}_{t\in\mathbb{R}}$ is a $w^{*}$-continuous action of $\mathbb{R}$ on $\mathcal{L}(\mathcal{F}(n,m,u))$ that normalizes both $\mathcal{A}_{u}$ and $\mathcal{L}_{u}$ and

[TABLE]

for all $a\in\mathcal{L}(\mathcal{F}(n,m,u))$. Then $\Phi_{k}$ leaves $\mathcal{L}_{u}$ invariant.

We can define the algebra $\mathcal{R}_{u}$ generated by the right shifts $R_{e}$ and $R_{f}$ defined by

[TABLE]

and

[TABLE]

The techniques of the proof of Proposition 2.3 of [5] can be applied here to show that the commutant of $\mathcal{R}_{u}$ is $\mathcal{L}_{u}$. Also, mapping $e_{i_{1}}\otimes e_{i_{2}}\otimes\cdots\otimes e_{i_{k}}\otimes f_{j_{1}}\otimes f_{j_{2}}\otimes\cdots\otimes f_{j_{l}}$ to $f_{j_{l}}\otimes f_{j_{l-1}}\otimes\cdots\otimes f_{j_{1}}\otimes e_{i_{k}}\otimes e_{i_{k-1}}\otimes\cdots\otimes e_{i_{1}}$, we get a unitary operator

[TABLE]

implementing a unitary equivalence of $\mathcal{L}_{u}$ and $\mathcal{R}_{u^{*}}$. In fact, it is easy to check that $R_{e_{i}}W=WL_{e_{i}}$ and $R_{f_{j}}W=WL_{f_{j}}$ for every $i,j$. To see that the commutation relation in the range is given by $u^{*}$, apply $W$ to (2) to get (in the range of $W$) $e_{k}\otimes f_{l}=\sum_{i,j}\bar{u}_{(i,j),(k,l)}f_{j}\otimes e_{i}=\sum_{i,j}(u^{*})_{(k,l),(i,j)}f_{j}\otimes e_{i}$ which is equation (1) with $u^{*}$ instead of $u$.

As in [5], we conclude that $(\mathcal{L}_{u})^{\prime}=\mathcal{R}_{u}$ and $(\mathcal{L}_{u})^{\prime\prime}=\mathcal{L}_{u}$.

3 The character space and its core

In the following proposition we describe the structure of the character spaces $\mathcal{M}(\mathcal{L}_{u})$ and $\mathcal{M}(\mathcal{A}_{u})$ (equipped with the weak∗ topology). Similar results were obtained in [5] for algebras defined for higher rank graphs and in [2] for analytic Toeplitz algebras. (See also [10].)

It will be convenient to write

[TABLE]

and

[TABLE]

where $\mathbb{B}_{n}$ is the open unit ball of $\mathbb{C}^{n}$. We refer to $V_{u}$ as the *variety associated with * $u$.

Proposition 3.1

1. (1)

The linear multiplicative functionals on $\mathbb{C}[\mathbb{F}^{+}_{u}]$ are in one-to-one correspondence with points $(z,w)$ in $V_{u}$. 2. (2)

$\mathcal{M}({\mathcal{A}}_u)$