Kumjian, Alexander Preliminary algebras arising from local homeomorphisms. (English) Zbl 0541.46047 Math. Scand. 52, 269-278 (1983). Let \(\psi\) :\(X\to T\) be a continuous open surjection. \(C_ c(X)\) denotes the set of all continuous functions with compact support. The author proved that \(\ell^ 2(\psi)\) is the completion of \(C_ c(X)\) for the norm induced by a suitable inner \(product.\) \(C^*(\psi)\) denotes the imprimitivity algebra (viz. the closed *- subalgebra of \(End(\ell^ 2(\psi))\) generated by operators of the form \(<f,g> (f,g\in \ell^ 2(\psi))\), where \(<f,g>h=f(g| h)\) for each \(h\in \ell^ 2(\psi).\) The main object of this paper is to study the algebra \(C^*(\psi)\). The results are as follows: (1) The group of automorphisms of \(C^*(\psi)\) fixing the diagonal (i.e. \(\alpha {\mathbb{O}}p=p)\) is isomorphic to \(Z'(R(\psi),T).\) (2) Let \(\delta\) be a *-derivation on \(C_ c(R(\psi))\) so that \(\delta {\mathbb{O}}p=0\). There is \(d\in Z'(R(\psi),R)\) such that \(\delta f=id\cdot f.\) Further, \(\exists p:X\to R\) continuous so that \(d(x,y)=p(x)-p(y)\) and the one-parameter automorphism group is inner. Reviewer: Rho Jae-Chul Cited in 10 Documents MSC: 46L05 General theory of \(C^*\)-algebras 46L40 Automorphisms of selfadjoint operator algebras 46L10 General theory of von Neumann algebras 46K10 Representations of topological algebras with involution 43A07 Means on groups, semigroups, etc.; amenable groups Keywords:faithful conditional expectation; *-derivation; imprimitivity algebra; one-parameter automorphism group PDFBibTeX XMLCite \textit{A. Kumjian}, Math. Scand. 52, 269--278 (1983; Zbl 0541.46047) Full Text: DOI EuDML