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- W2907172647 abstract "Necessary and sufficient conditions are given in terms of E' that a weak topology w(E, E') on an algebra E be Aconvex. The main condition is that each element g of E' contain a weakly closed subspace L of finite codimension such that g is bounded on all multiplicative translates of L. For weak topologies, A-convexity (which assumes only separate continuity of multiplication) is equivalent to joint continuity of multiplication. Let E be an algebra, E' a total subspace of the dual of E and w(E, E') the weak topology of E determined by E'. The purpose of this paper is to determine necessary and sufficient conditions that w(E, E') be A-convex. Warner [4] has given a necessary and sufficient condition that w(E, E') be locally m-convex and also a necessary and sufficient condition that multiplication be jointly (w(E, E')) continuous. One of the equivalent forms of our condition is that w(E, E') is A-convex (which requires only the separate continuity of multiplication) if and only if multiplication is jointly (w(E, E')) continuous. Thus, all weak topological algebras (joint continuity of multiplication) are already A-convex. A-convex algebras, which include the locally m-convex algebras, were introduced in [2]. In ?2 the basic properties are given along with some examples. The main results are given in ?3. 2. A-convex algebras. Throughout this note, E will denote an algebra, E' a total subspace of the dual of E and w(E, E') the weak topology on E induced by E'. The proofs of the results given here maybe found in [2 ]. (2.1) DEFINITION. A subset V of E is called A-convex if V is absolutely convex, absorbing and for each xCE, V absorbs x V and Vx. The inverse image of an A-convex set under a homomorphism is A-convex, as is the image of an A-convex set under a surjective homomorphism. (2.2) DEFINITION. An A-convex algebra is an algebra E together with a topology on E whose neighborhood system at zero has a basis of A-convex sets. Presented to the Society, January 23, 1970; received by the editors October 3, 1969. AMS 1968 subject classifications. Primary 4650; Secondary 4601, 4625, 1620." @default.
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- W2907172647 date "1970-01-01" @default.
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- W2907172647 title "Weak $A$-convex algebras" @default.
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- W2907172647 doi "https://doi.org/10.1090/s0002-9939-1970-0262830-x" @default.
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