On locally \(M\)-convex \(\ast\)-algebra bundles
The notion of topological algebra bundle has already been given by \textit{A. Kyriazis} in the paper [``On topological algebra bundles'', J. Aust. Math. Soc., Ser. A 43, 398-419 (1987; Zbl 0672.46045)]. Briefly, a topological algebra bundle over a Hausdorff topological space \(X\) is a triple \(\xi= (E,\pi, X)\) with \(E\) a nonempty topological space and \(\pi: E\to X\) a continuous map whose fibre \(E_x= \pi^{-1}(x)\), \(x\in X\), is a topological algebra in such a way that the standard local triviality conditions are satisfied. In particular, a topological \(*\)-algebra bundle is a topological algebra bundle whose fibres are topological \(*\)-algebras. For \(\xi\) a locally \(m\)-convex (lmc) \(*\)-algebra bundle, the authors define here the enveloping locally \(C^*\)-algebra bundle \({\mathcal E}(\xi)\) where the fibres are the enveloping locally \(C^*\)-algebra. So, the enveloping functor \({\mathcal E}\) is obtained (Theorem 3.1). Moreover, related to the sections functor \(\Gamma\), this functor satisfies (under certain conditions) \({\mathcal E}(\Gamma(\xi))= \Gamma({\mathcal E}(\xi))\) with respect to an isomorphism of locally \(C^*\)-algebras. In the last section, as an application of the functor \({\mathcal E}\) an analogue of \textit{A. Mallios'} result [``Topological algebras, selected topics'', North-Holland, Amsterdam (1986; Zbl 0597.46046), 6.4, p. 498] is obtained in the case of an lmc \(C^\infty\)-\(*\)-algebra bundle \(\xi\).
locally \(M\)-convex \(*\)-algebra bundles
scientific article; zbMATH DE number 1393013
scientific article