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Preliminary algebras arising from local homeomorphisms. (English) Zbl 0541.46047

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

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
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