Daniil Proskurin

Stability of a Special Class of \(q_{ij}\)-CCR and Extensions of Higher-Dimensional Noncommutative Tori

AbstractThe

Operator Relations, Dynamical Systems, and Representations of a Class of Wick Algebras

\mathop C\nolimits^*

On *-representations of a deformation of a wick analogue of the car algebra

-algebras A{qi}, Θ generated by generalised quon commutation relations are considered. The nuclearity of these algebras is proved. It is shown that A{qi}, Θ is isomorphic to the extension of a higher-dimensional noncommutative torus. Irreducible representations of A{qi}, Θ are considered. It is shown that the Fock representation is faithful.

On structure of homogenenous Wick ideals in Wick *-algebras with braided coefficients

We consider the C*-algebras and generated, respectively, by isometries s1, s2 satisfying the relation s*1s2 = qs2s*1 with |q| < 1 (the deformed Cuntz relation), and by isometries s1, s2 satisfying the relation s2s1 = qs1s2 with |q| = 1. We show that is isomorphic to the Cuntz–Toeplitz C*-algebra for any |q| < 1. We further prove that if and only if either q1 = q2 or . In the second part of our paper, we discuss the complexity of the representation theory of . We show that is *-wild for any q in the circle |q| = 1, and hence that is not nuclear for any q in the circle.

A Family of *-Algebras Allowing Wick Ordering: Fock Representations and Universal Enveloping C*-Algebras

We consider families of operators satisfying a general class of relations, whose solutions can be described in terms of orbits of some dynamical system acting on the spectrum of a commuting sub-family. In the first section we introduce a class of relations and show, how the representations of such relations are related to orbits of the corresponding dynamical system. Also, we discuss the problem of accurate sense of the relation for unbounded operators. In Section 2, we study the class of *-algebras allowing Wick ordering whose representations can be studied by using methods of Section 1. We classify such Wick *-algebras, and discuss their representations.

The kernel of Fock representations of Wick algebras with braided operator of coefficients

We study a deformation of the Cuntz–Toeplitz C*-algebra determined by the relations

On C -algebras generated by some deformations of CAR relations

{a_i^*a_i=1+q a_ia_i^*,\, a_i^*a_j=0}