On C*-Algebras Generated by Isometries with Twisted Commutation Relations

TL;DR

This paper investigates the structure and properties of C*-algebras generated by isometries with twisted commutation relations, revealing differences in nuclearity and exactness, and computing their K-theory, thus addressing a longstanding question in the field.

Contribution

It introduces and analyzes a new class of C*-algebras generated by isometries with twisted relations, highlighting their non-exactness and computing their K-groups.

Findings

The tensor product case C*-algebra is nuclear.

The weaker case C*-algebra is not exact.

Both cases have K_0 = Z and K_1 = 0.

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.