Shilov boundary for "holomorphic functions" on a quantum matrix ball

We describe the Shilov boundary ideal for a q-analog of algebra of holomorphic functions on the unit ball in the space of $2\times 2$ matrices.Comment: 14 page

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems

On C*-algebras generated by pairs of q-commuting isometries

We consider the C*-algebras O_2^q and A_2^q generated, respectively, by isometries s_1, s_2 satisfying the relation s_1^* s_2 = q s_2 s_1^* with |q| < 1 (the deformed Cuntz relation), and by isometries s_1, s_2 satisfying the relation s_2 s_1 = q s_1 s_2 with |q| = 1. We show that O_2^q is isomorphic to the Cuntz-Toeplitz C*-algebra O_2^0 for any |q| < 1. We further prove that A_2^{q_1} is isomorphic to A_2^{q_2} if and only if either q_1 = q_2 or q_1 = complex conjugate of q_2. In the second part of our paper, we discuss the complexity of the representation theory of A_2^q. We show that A_2^q is *-wild for any q in the circle |q| = 1, and hence that A_2^q is not nuclear for any q in the circle.Comment: 18 pages, LaTeX2e "article" document class; submitted. V2 clarifies the relationships between the various deformation systems treate

Unbounded representations of qq-deformation of Cuntz algebra

We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence.Comment: 13 pages, Submitted to Lett. Math. Phy