Yu. S. Samoilenko

For a $C^*$-algebra generated by a finite family of isometries $s_j$, $j=1,\dots,d$, satisfying the $q_{ij}$-commutation relations \[ s_j^* s_j = I, \quad s_j^* s_k = q_{ij}s_ks_j^*, \qquad q_{ij} = \bar q_{ji}, |q_{ij}|<1, \ 1\le i \ne j \le d, \] we construct an infinite family of unitarily non-equivalent irreducible representations. These representations are deformations of a corresponding class of representations of the Cuntz algebra $\mathcal O_d$.

Для $C^*$-алгебри, породженої скінченною сім’єю ізометрій $s_j$, $j=1,\dots,d$, що задовольняє $q_{ij}$-комутаційним співвідношенням \[ s_j^* s_j = I, \quad s_j^* s_k = q_{ij}s_ks_j^*, \qquad q_{ij} = \bar q_{ji}, |q_{ij}|<1, \ 1\le i \ne j \le d, \] ми будуємо нескінченну сім'ю унітарно нееквівалентних незвідних представлень. Ці представлення є деформаціями відповідного класу представлень алгебри Кунца $\mathcal O_d$.

In this paper, we find $\tau$, $0<\tau<1$, such that there exists an equiangular $(\Gamma, \tau)$-configuration of one-dimensional subspaces, and describe $(\Gamma, \tau)$-configurations that correspond to unicyclic graphs and to some graphs that have cyclomatic number satisfying $\nu(\Gamma) \geq 2$.

We study systems of one-dimensional subspaces of a Hilbert space. For such systems, symmetric and orthoscalar systems, as well as graph related configurations of one-dimensional subspaces have been studied.

We study a relation between brick $n$-tuples of subspaces of a finite dimensional linear space, and irreducible $n$-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive coefficients, of orthogonal projections onto these subspaces equals the identity operator. We prove that brick systems of one-dimensional subspaces and the systems obtained from them by applying the Coxeter functors (in particular, all brick triples and quadruples of subspaces) can be unitarized. For each brick triple and quadruple of subspaces, we describe sets of characters that admit a unitarization.

We consider the complexity of the representation theory of free products of $C^*$-algebras. Necessary and sufficient conditions for the free product of finite-dimensional $C^*$-algebras to be $*$-wild is presented. As a corollary we get criteria for $*$-wildness of free products of finite groups. It is proved that the free product of a non-commutative nuclear $C^*$-algebra and the algebra of continuous functions on the one-dimensional sphere is $*$-wild. This result is applied to estimate the complexity of the representation theory of certain $C^*$-algebras generated by isometries and partial isometries.

In the present work a relationship between systems of n subspaces and representations of *-algebras generated by projections is investigated. It is proved that irreducible nonequivalent *-representations of *-algebras P4,com generate all nonisomorphic transitive quadruples of subspaces of a finite dimensional space.