On C ∗-algebras generated by isometries with twisted commutation relations

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 ⊗ϑ T 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 ∗ϑ T 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 ⊗ϑ T . We show that T ⊗ϑ T is nuclear, whereas T ∗ϑ T is not even exact. Also, we compute the K-groups and we obtain K0(T ∗ϑ T ) = Z and K1(T ∗ϑ T ) = 0, and the same K-groups for T ⊗ϑ T . © 2013 Elsevier Inc. All rights reserved.