SEMICROSSED PRODUCTS OF C*-ALGEBRAS AND THEIR C*-ENVELOPES

Let C be a C*-algebra and alpha: C -> C a unital *- endomorphism. There is a natural way to construct operator algebras, called semicrossed products, using a convolution induced by the action of alpha on C. We show that the C*-envelope of a semicrossed product is (a full corner of) a crossed product. As a consequence, we get that when alpha is *-injective, the semicrossed products are completely isometrically isomorphic and share the same C*-envelope, the crossed product C-infinity (sic)(alpha infinity) Z. We show that minimality of the dynamical system (C, alpha) is equivalent to non-existence of non-trivial Fourier invariant ideals in the C*-envelope. We get sharper results for commutative dynamical systems.