A new look at the crossed-product of a C^*-algebra by an endomorphism

Abstract

We give a new definition for the crossed-product of a C^*-algebra A by a ^*-endomorphism $\alpha$, which depends not only on the pair $(A,\alpha)$ but also on the choice of a transfer operator. With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz–Krieger algebra $\mathcal{O}_{\mathcal A}$ arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transfer operator.