V. L. Ostrovskyi

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Articles: 8

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 $\mathcal O_d$.

Some remarks on Hilbert representations of posets

V. Ostrovskyi, S. Rabanovich

↓ Abstract   |   Article (.pdf)

MFAT 20 (2014), no. 2, 149–163


For a certain class of finite posets, we prove that all their irreducible orthoscalar representations are finite-dimensional and describe those, for which there exist essential (non-degenerate) irreducible orthoscalar representations.

Representations of relations with orthogonality condition and their deformations

V. L. Ostrovskyi, D. P. Proskurin, R. Y. Yakymiv

↓ Abstract   |   Article (.pdf)

MFAT 18 (2012), no. 4, 373-386


Irreducible representations of $*$-algebras $A_q$ generated by relations of the form $a_i^*a_i+a_ia_i^*=1$, $i=1,2$, $a_1^*a_2=qa_2a_1^*$, where $q\in (0,1)$ is fixed, are classified up to the unitary equivalence. The case $q=0$ is considered separately. It is shown that the $C^*$-algebras $\mathcal{A}_q^F$ and $\mathcal{A}_0^F$ generated by operators of Fock representations of $A_q$ and $A_0$ are isomorphic for any $q\in (0,1)$. A realisation of the universal $C^*$-algebra $\mathcal{A}_0$ generated by $A_0$ as an algebra of continuous operator-valued functions is given.

On quadruples of linearly connected projections and transitive systems of subspaces

Yulia Moskaleva, Vasyl Ostrovskyi, Kostyantyn Yusenko

↓ Abstract   |   Article (.pdf)

MFAT 13 (2007), no. 1, 43-49


We study conditions under which the images of irreducible quadruples of linearly connected projections give rise to all transitive systems of subspaces in a finite dimensional Hilbert space.

On $*$-representations of a certain class of algebras related to a graph

Vasyl Ostrovskyi

MFAT 11 (2005), no. 3, 250-256


Centered one-parameter semigroups

Vasyl Ostrovskyi

MFAT 10 (2004), no. 2, 32-42


On double commutator relation

Vasyl Ostrovskyĭ

MFAT 6 (2000), no. 2, 60-65


On operator relations, centered operators, and nonbijective dynamical systems

Vasyl’ Ostrovs’kyj

MFAT 2 (1996), no. 3, 114-121


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