Representations of canonical anticommutation relations with orthogonality condition

Abstract

We study the class of Hilbert space representations of the ∗-algebra $A^{(d)}_0$ generated by relations of the form $$A^{(d)}_0 = \mathbb{C}\langle a_j, a_j^{*} | a_j^{*} a_j = 1 - a_j a_j^{*},\; a_j, a_j^{*} = 0, i \neq j,\; i, j = 1,...,d\rangle,$$ Namely, we describe the classes of unitary equivalence of irreducible representations of $A^{(d)}_0$ such that there exists $j = 1,...,d$ for which $a^2_j \neq 0$.