On C~*-algebras generated by isometries with twisted commutation relations

摘要

In the theory of C~*-algebras, interesting noncommutative structures arise as deformations of the tensor product, e.g. the rotation algebra A_(θ{symbol}) 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{symbol}T generated by two isometries u and v such that uv=e2πiθ{symbol}vu and u*v=e-2πiθ{symbol}vu~*, for θ{symbol}∈R. Since the second relation implies the first one, we also consider the universal C~*-algebra T~*θ{symbol}T generated by two isometries u and v with the weaker relation uv=e2πiθ{symbol}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" T?θ{symbol}T. We show that T?θ{symbol}T is nuclear, whereas T*θ{symbol}T is not even exact. Also, we compute the K-groups and we obtain K0(T*θ{symbol}T)=Z and K1(T*θ{symbol}T)=0, and the same K-groups for T?θ{symbol}T.