On functional representations of topological algebras

Abstract

The category of topological algebras we are concerned with is that of m-barreled ones for which the underlying topological vector space is not necessarily locally convex. These algebras specialize to Fréchet locally m-convex (topological) algebras (Michael) or to (locally m-convex) inductive limits of such (Warner) and they are characterized from being such that every balanced, convex, closed, idempotent and absorbing set ( m-barrel) in such an algebra to be a local neighborhood. For a topological algebra of this kind, the equicontinuous sets of its spectrum, the weakly relatively compact sets and the weakly bounded sets are the same (Corollary 2.1). As a consequence, in case of a commutative, advertibly complete, m-barreled locally m-convex algebra, equicontinuity of its spectrum is equivalent with the algebra being bounded, or even a Q-algebra, or its spectrum a weakly relatively compact subset of the respective topological dual space (Corollary 2.4). This specializes to a previous result of E. A. Michael. Now, for an m-barreled topological algebra E, whose the underlying topological vector space is a Pták locally convex one, the algebraic exactness of the sequence ▪ where g is the respective Gel'fand map, implies the sequence to be also topologically exact (: g is a topological isomorphism) in such a way that the spectrum of E is a k-space and the topology of the algebra is that of the uniform convergence on the closed equicontinuous subsets of its spectrum (Theorem 3.1). For Fréchet algebras this yields a result of S. Warner (Corollary 3.1), the spectrum of the algebra being, moreover, in this case a (Hausdorff completely regular) hemicompact k-space. Further information, but of a more technical nature, in case of a commutative, semi-simple, m-barreled topological algebra concerning a situation resembling that of the classical Gel'fand-Neumark representation theorem is provided by Theorem 3.2 and its Corollary. This extends also previous results of P. D. Morris and D. E. Wulbert. The last section contains material indicating that local equicontinuity of the spectrum of a topological algebra is not always a necessary condition as it concerns some basic formulas relating the spectrum of a topological tensor product algebra to the spectra of the factor algebras (Theorem 4.1).