On C*-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. For instance, the rotation algebra may be seen as a scalar twist deformation of the tensor product of the functions on the circle with itself. We deform the tensor product of two Toeplitz algebras in the same way, introducing the universal C*-algebra generated by two isometries u and v such that uv=e^{it}vu and u*v=e^{-it}vu*, for a fixed real parameter t. Since the second relation implies the first one, we also consider the universal C*-algebra generated by two isometries u and v with the weaker relation uv=e^{it}vu. 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" of two Toeplitz algebras. We show that the C*-algebra in the "tensor product case" is nuclear, whereas in the "weaker case" it is not even exact. Also, we compute the K-groups and we obtain K_0 = Z and K_1 = 0 for both C*-algebras. This answers a question raised by Murphy in 1994 concerning the K-theory of the C*-algebra associated to N^2.