Proskurin D. Deformations of CCR: representations and enveloping C*-algebras

The thesis is devoted to the study of deformations of canonical commutation and anti-commutation relations and their generalisations. Namely the algebra generated by twisted CCR, generalized quons algebra, Wick algebras with braided coefficients and Wick analogues of CCR and CAR are considered. It is proved that Fock representations of C*-algebras generated by twisted CCR, generalized quons C*-algebra and of q-version of polinomials over unit disk in tha space of complex 2 by 2 matrices are faithful. It is shown that under any admissible values of deformation parameter C*-algebras generated by twisted CCR are pairwise isomorphic. It is proved that if operator of coefficients of Wick algebra s braided and its norm is less than one, then the C*-algebra, generated by Fock representation contains the Cuntz-Toeplitz algebra. The structue of homogeneous Wick ideals in algebras with braided coefficients is studied. It is shown that tensor realisation of monotone and boolean independence is universal. The notion of *-wild algebra is generalized to the case of algebra without finite-dimensional representations.