PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 2, December 1973 REPRESENTATION OF ^-CONVEX ALGEBRAS ALLAN C. COCHRAN Abstract. Algebraic properties of A-convex algebras are developed via a functor to locally m-convex algebras. The Gel'fand- Mazur theorem holds for A-convex algebras, and this fact allows a Gel'fand-type representation theorem for a subclass of uniformly ^(-convex algebras. Connections to existing functional represen- tation theory are also obtained. 1. Introduction. The objects of this note are to give some algebraic properties of ^-convex algebras, to note the fact that the Gel'fand-Mazur theorem holds for this class of algebras and to develop a Gel'fand-type theorem appropriate for certain /á-convex algebras. The main technique for obtaining algebraic properties is to use a functor from the category of yl-convex algebras to the category of locally m-convex algebras and then use the existing theory of m-convex algebras (see, for example, [5]). Various relations are developed along these lines. The fact that the Gel'fand-Mazur theorem holds for A-convex algebras means that for commutative algebras there is a one-to-one correspondence between continuous nonzero multiplicative linear functionals and the closed regular maximal ideals. However, the generalizations of the Gel'fand representation theory on this carrier space (e.g. [6]) are all in terms of a locally m-convex algebra. We develop a representation theorem for a subclass of y4-convex algebras (uniformly ^4-convex) which gives the m-convex results as corollaries. A type of strict topology is used on the space of continuous functions in which the algebra is embedded. Basic results on A-convex algebras are found in [1], [2] and [3]. In order to make this note relatively self-contained we briefly repeat pertinent definitions. A locally convex algebra is an algebra over R or C with a locally convex topology for which multiplication is separately continuous. An A-convex seminorm on an algebra F is a seminorm p such that for x e E there are constants Mx and Nx such that p(xy) = Mxp(y) andp(yx)^ Nxp(y) for all y e E. An A-convex algebra {locally m-convex algebra} Received by the editors March 15, 1973. AMS (MOS) subject classifications (1970). Primary 46H15; Secondary '6H05, 46H20. Key words and phrases, ^-convex algebra, locally m-convex algebra, Gel'fand-Mazur theorem, strict topology, compact-open topology, Gel'fand representation theorem. © American Mathematical Society 1973 473 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use