Yurii Samoilenko

THE SPECTRAL PROBLEM AND *-REPRESENTATIONS OF ALGEBRAS ASSOCIATED WITH DYNKIN GRAPHS

We study the connection between *-representations of algebras associated with graphs, locally-scalar graph representations and the problem about the spectrum of a sum of two Hermitian operators. For algebras associated with Dynkin graphs we give an explicit description of the parameters for which there are irreducible representations and an algorithm for constructing these representations.

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

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.

Representations of posets: linear versus unitary

A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially ordered sets in linear spaces. It happens that the results in the unitary case are well-correlated with those in the linear case. The purpose of this article is to shed light on this phenomena.

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

We consider an abstract Wick ordering as a family of relations on elements ae and define *-algebras by these relations. The relations are given by a fixed operator \( T:\mathfrak{h} \oplus \mathfrak{h} \to \mathfrak{h} \oplus \mathfrak{h} \), where \( \mathfrak{h} \) is one-particle space, and they naturally define both a *-algebra and an inner-product space H T , 〈·,·〉 T . If a i * denotes the adjoint, i.e., a i * , then we identify when 〈·,·〉 T is positive semidefinite (the positivity question!). In the case of deformations of the CCR-relations (the qij-CCR and the twisted CCR’s), we work out the universal C* -algebras \( \mathfrak{A} \), and we prove that, in these cases, the Fock representations of the \( \mathfrak{A} \) are faithful.