  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 (defined in the paper). 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 O_A arises as the result of our
construction when applied to the corresponding Markov subshift and a very
natural transfer operator.
