On the generalized spectra of topological algebras

Abstract

Local equicontinuity of the generalized spectra of certain topological algebras implies that the spectra in question coincide as topological spaces with the generalized spectra of the respective completed topological algebras [7]. On the other hand, for a fairly large class of topological algebras (i.e., m-barreled locally convex algebras), local equicontinuity of the (usual) spectrum is equivalent to its local compactness [6]. In this paper we are concerned with generalized spectra of (nonnecessarily commutative [II]) topological algebras. The results obtained have a special bearing, for the case under consideration, on the corresponding results contained in [6, 7, 1 I], which were also the motivation to the present setting. Thus, let E and F be topological algebras and let A(E, F) be the generalized spectrum of E (for F given) (i.e., the set of nonsero continuous algebra homomorphisms of E into F topologized as a subset of P8(E, F); cf. Section 3 below). It is shown that if E is an m-barreled locally convex algebra and F a locally m-convex semiMonte1 one, local equicontinuity of d(E, F) is equivalent to its local compactness (cf., Theorem 3.2). Furthermore, we are dealing with representations of the generalized spectra of topological tensor product algebras. In this respect, it is proved that there exists a bicontinuous injection of the generalized spectrum of a locally convex infinite tensor product algebra into the Cartesian product space of the spectra of the factor algebras (cf., Theorem 5.2). Moreover, by considering completed topological algebras, we still assume local equicontinuity for the generalized spectra. This assumption yields analogous representations of the generalized spectra of certain completed locally convex infinite tensor product algebras (with continuous multiplication) (Theorem 5.3). On the other hand, with the class of m-barreled locally convex algebras several aspects concerning the continuity of the Gel’fand map are closely related [6]. Thus, we are also interested therein in the continuity of the generalized Gel’fand map (cf. Section 4). The results obtained constitute extended forms of those of [6; Section 31.