On C*-algebras generated by pairs of q-commuting isometries

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We consider the C*-algebras O₂q and A₂q generated, respectively, by isometries s₁, s₂ satisfying the relation s₁* s₂ = q s₂ s₁* with |q| < 1 (the deformed Cuntz relation), and by isometries s₁, s₂ satisfying the relation s₂ s₁ = q s₁ s₂ with |q| = 1. We show that O₂q is isomorphic to the Cuntz-Toeplitz C*-algebra O₂⁰ for any |q| < 1. We further prove that A₂q¹ is isomorphic to A₂q² if and only if either q₁ = q₂ or q₁ = complex conjugate of q₂. In the second part of our paper, we discuss the complexity of the representation theory of A₂q. We show that A₂q is *-wild for any q in the circle |q| = 1, and hence that A₂q is not nuclear for any q in the circle.