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

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....

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Published in:Ergodic theory and dynamical systems 2003-12, Vol.23 (6), p.1733-1750, Article S0143385702001797
Main Author: EXEL, RUY
Format: Article
Language:English
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Summary: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385702001797