On [formula omitted]-algebras generated by isometries with twisted commutation relations

Abstract

In the theory of C⁎-algebras, interesting noncommutative structures arise as deformations of the tensor product, e.g. the rotation algebra Aϑ as a deformation of C(S1)⊗C(S1). We deform the tensor product of two Toeplitz algebras in the same way and study the universal C⁎-algebra T⊗ϑT generated by two isometries u and v such that uv=e2πiϑvu and u⁎v=e−2πiϑvu⁎, for ϑ∈R. Since the second relation implies the first one, we also consider the universal C⁎-algebra T⁎ϑT generated by two isometries u and v with the weaker relation uv=e2πiϑvu. Such a “weaker case” does not exist in the case of unitaries, and it turns out to be much more interesting than the twisted “tensor product case” T⊗ϑT. We show that T⊗ϑT is nuclear, whereas T⁎ϑT is not even exact. Also, we compute the K-groups and we obtain K0(T⁎ϑT)=Z and K1(T⁎ϑT)=0, and the same K-groups for T⊗ϑT.