R. Y. Yakymiv

We consider an $C^*$-algebra $\mathcal{E}_{1,n}^q$, $q\le 1$, generated by isometries satisfying $q$-deformed commutation relations. For the case $|q|<1$, we prove that $\mathcal E_{1,n}^q \simeq\mathcal E_{1,n}^0=\mathcal O_{n+1}^0$. For $|q|=1$ we show that $\mathcal E_{1,n}^q$ is nuclear and prove that its Fock representation is faithul. In this case we also discuss the representation theory, in particular construct a commutative model for representations.

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.

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.