D. P. Proskurin
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On well-behaved representations of $\lambda$-deformed CCR
D. P. Proskurin, L. B. Turowska, R. Y. Yakymiv
Methods Funct. Anal. Topology 23 (2017), no. 2, 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
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