Green's result and Morita equivalence

Question

This question is from this note, 7.1.2, it says "by a result of Green", but I do not know how to get it.

Let $A$ be a C$^*$-algebra, $\alpha\in \text{Aut}(A)$, define $$M_{\alpha} = \{f\in C(M,\mathbb R):\ f(1)=\alpha(f(0))\}$$ (mapping torus).

We have $\mathbb R$ acting on $M_{\alpha}$ by $(\beta_tf)(s)= f(s-t)$. Prove that $A\times_{\alpha}\mathbb Z$ is equivalent to $M_{\alpha}\times_{\beta}\mathbb R$.

Answer 1

I think you mean to take functions with values in $A$. This document can help you, look at page 23. Basically, the $\mathbb{Z}$-action groupoid is obtained by relativizing the $\mathbb{R}$-action groupoid to a transversal, this procedure results in an equivalent groupoid (in the sense of Renault).