On the spectra of topological algebras

Abstract

Equicontinuity of the spectra of topological algebras was proved to be of a particular significance in connection with topological tensor product algebras, especially when (topological) completeness of such algebras had to be considered (cf., for instance, [II]. [I7]). On the other hand, it was also proved [13] that only local equicontinuity [3] is needed, whenever, in this respect, equicontinuity has been used instead. We consider in this paper a fairly large class of topological algebras, i.e., m-barreled locally convex algebras (cf. Section 2), for which local equicontinuity of the spectrum is equivalent to its local compactness. FrCchet algebras with locally compact spectrum, e.g., Stein algebras (cf. Section 4 below), and, more generally, topological inductive limits of such algebras belong to this category of algebras. The situation described comprises a recent result in ([3]; p. 108, Theorem 3.1), which was also the motivation to the present setting. We consider, furthermore, an application of the preceding concerning the spectrumof certain infinite tensor product topological algebras [12]. An other application of the said result in [3] combined with the theory of topological tensor product algebras [II] is given by considering topological tensor products of Stein algebras (cf. Section 4 below). Finally, a particular class of the topological algebras dealt with in the following has also been recently considered in [IL?]. Thus, Section 3 below gives information concerning the continuity of the Gel’fand map for the topological algebras involved, which extend and improve the corresponding results in [ZS].