# D. P. Proskurin

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### Representations of relations with orthogonality condition and their deformations

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

Methods Funct. Anal. Topology **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 $C^*$-algebra generated by Fock representation of Wick algebra with braided coefficients

Methods Funct. Anal. Topology **17** (2011), no. 2, 168-173

We consider $C^*$-algebras $\mathcal{W}(T)$ generated by operators of Fock representations of Wick $*$-algebras with a braided coefficient operator $T$. It is shown that for any braided $T$ with $||T||<1$ one has the inclusion $\mathcal{W}(0)\subset\mathcal{W}(T)$. Conditions for existence of an isomorphism $\mathcal{W}(T)\simeq\mathcal{W}(0)$ are discussed.

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

Sergio Albeverio, Kate Jushenko, Daniil Proskurin, Yurii Samoilenko

Methods Funct. Anal. Topology **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.

### On the monotone independent families of operators

Methods Funct. Anal. Topology **10** (2004), no. 2, 64-68

### On C*-algebra associated with ${\rm Pol}({\rm Mat}_ {2,2})_q$

Daniil Proskurin, Lyudmila Turowska

Methods Funct. Anal. Topology **7** (2001), no. 1, 88-92

### Stability of special class of $q_{ij}$-CCR and extensions of irrational rotation algebras

Methods Funct. Anal. Topology **6** (2000), no. 3, 97-104

### Representations of Wick CCR algebra

Methods Funct. Anal. Topology **5** (1999), no. 2, 83-85

### About positivity of Fock inner product of a certain Wick algebras

Methods Funct. Anal. Topology **5** (1999), no. 1, 88-94