# D. P. Proskurin

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Articles: 10

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

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

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 well-behaved representations of $\lambda$-deformed CCR

MFAT 23 (2017), no. 2, 192-205

192-205

We study well-behaved ∗-representations of a λ-deformation of Wick analog of CCR algebra. Homogeneous Wick ideals of degrees two and three are described. Well-behaved irreducible ∗-representations of quotients by these ideals are classified up to unitary equivalence.

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

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

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

D. Proskurin

MFAT 17 (2011), no. 2, 168-173

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

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

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

MFAT 10 (2004), no. 2, 64-68

64-68

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

MFAT 7 (2001), no. 1, 88-92

88-92

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

MFAT 6 (2000), no. 3, 97-104

97-104

### Representations of Wick CCR algebra

MFAT 5 (1999), no. 2, 83-85

83-85

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

MFAT 5 (1999), no. 1, 88-94

88-94