Semi-crossed products of C*-algebras
Let A be a C *-algebra and \(\alpha\) a *-endomorphism of A. The author defines the semicrossed product \({\mathbb{Z}}\) \(+\times_{\alpha}A\) of A by \(\alpha\) as the completion of the (noninvolutive) convolution algebra \(\ell\) 1(\({\mathbb{Z}}\) \(+,A,\alpha)\) in a certain operator enveloping norm. Semicrossed products include as special cases certain nonselfadjoint subalgebras of C *-crossed products, of which the Arveson-Josephson algebra [\textit{W. B. Arveson} and \textit{K. B. Josephson}, same journal 4, 100-134 (1969; Zbl 0186.459)] are particular instances. [For a similar construction in the W *-algebra setting, see, e.g., the paper of \textit{M. McAsey}, \textit{P. S. Muhly} and \textit{K.-S. Saito}, J. Math. Soc. Japan 33, No.3, 485-495 (1981; Zbl 0469.46046)]. The main results of this article describe semicrossed products of the form \({\mathbb{Z}}\) \(+\times_{\alpha}C_ 0(S)\), where the endomorphism \(\alpha\) of the abelian C *-algebra \(A=C_ 0(S)\) is induced by a continuous proper mapping \(\phi\) of the locally compact space S. The author gives necessary and sufficient conditions on (S,\(\phi)\) in order that \({\mathbb{Z}}\) \(+\times_{\alpha}C_ 0(S)\) be (i) semiprime, (ii) semisimple, or (iii) strongly semisimple. Also, the strong structure space of \({\mathbb{Z}}\) \(+\times_{\alpha}C_ 0(S)\) is determined in two extreme cases. In the final section it is shown that two semicrossed products of the form \({\mathbb{Z}}\) \(+\times_{\alpha}C_ 0(S)\) are isomorphic if and only if the corresponding dynamical systems are conjugate, provided the underlying spaces are compact without periodic points.
scientific article; zbMATH DE number 4036505