Yu. S. Samoilenko

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Articles: 14

A class of representations of $C^*$-algebra generated by $q_{ij}$-commuting isometries

Olha Ostrovska, Vasyl Ostrovskyi, Danylo Proskurin, Yurii Samoilenko

↓ Abstract   |   Article (.pdf)

MFAT 28 (2022), no. 1, 89-94


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 $\mathcal O_d$.

On equiangular configurations of subspaces of a Hilbert space

Yu. S. Samoilenko, Yulia Yu. Yershova

↓ Abstract   |   Article (.pdf)

MFAT 17 (2011), no. 1, 84-96


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 $\nu(\Gamma) \geq 2$.

Systems of one-dimensional subspaces of a Hilbert space

R. V. Grushevoy, Yu. S. Samoilenko

↓ Abstract   |   Article (.pdf)

MFAT 16 (2010), no. 2, 131-139


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.

On $n$-tuples of subspaces in linear and unitary spaces

Yu. S. Samoilenko, D. Yu. Yakymenko

↓ Abstract   |   Article (.pdf)

MFAT 15 (2009), no. 1, 48-60


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.

$*$-wildness of some classes of $C^*$-algebras

Sergio Albeverio, Kate Jushenko, Daniil Proskurin, Yurii Samoilenko

↓ Abstract   |   Article (.pdf)

MFAT 12 (2006), no. 4, 315-325


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.

Systems of $n$ subspaces and representations of $*$-algebras generated by projections

Yu. P. Moskaleva, Yu. S. Samoĭlenko

↓ Abstract   |   Article (.pdf)

MFAT 12 (2006), no. 1, 57-73


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.

The spectral problem and algebras associated with extended Dynkin graphs. I.

S. A. Kruglyak, S. V. Popovych, Yuriĭ Samoĭlenko

MFAT 11 (2005), no. 4, 383-396


Spectral theorems for $*$-representations of the algebras $\mathcal{P}_{\Gamma,\chi,com}$ associated with Dynkin graphs

Yu. S. Samoǐlenko, M. V. Zavodovsky

MFAT 11 (2005), no. 1, 88-96


Von Neumann dimensions of symmetric and antisymmetric tensor products

Alexei Daletskii, Yuriĭ Samoĭlenko

MFAT 9 (2003), no. 2, 123-132


On "good" vectors for family of unbounded operators and their application

Yu. S. Samoĭlenko, A. V. Strelets

MFAT 8 (2002), no. 2, 88-100


On bounded and unbounded idempotents whose sum is a multiple of the identity

Yuriĭ Samoĭlenko, Lyudmila Turowska

MFAT 8 (2002), no. 1, 79-100


On the decomposition of the identity into a sum of idempotents

T. Ehrhardt, V. Rabanovich, Yu. Samoǐlenko, B. Silbermann

MFAT 7 (2001), no. 2, 1-6


On representations of $\mathcal F_n$-algebras and invertibility symbols

Slavik Rabanovich, Yuriĭ Samoĭlenko

MFAT 4 (1998), no. 4, 86-96


Semilinear relations and their $*$-representations

Yu. S. Samoĭlenko, V. S. Shulʹman, L. B. Turowska

MFAT 2 (1996), no. 1, 55-111


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