On the mapping torus of an automorphism

Abstract:

Let $\rho$ be an automorphism of a ${C^ * }$-algebra $A$. The mapping torus ${T_\rho }(A)$ is the ${C^ * }$-algebra of $A$-valued continuous functions $x$ on $[0,1]$ satisfying $x(1) = \rho (x(0))$. Using his Thom isomorphism theorem, A. Connes has shown that the $K$-groups of ${T_\rho }(A)$, with indices reversed, are isomorphic to those of the crossed product $A{ \times _\rho }Z$. We provide here an alternative proof of this fact which gives an explicit description of the isomorphism.