Completeness, Equicontinuity, and Hypocontinuity in Operator Algebras
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS I, 419-442 (1967) Completeness, Equicontinuity, and Hypocontinuity in Operator Algebras ROBERT T. MOORE* Department of Mathematics, University of California, Berkeley, California 94720 Communicated by Edward Nelson Received March 28. 1967 1. INTRODUCTION The results presented here provide three different types of operator- algebraic characterizations of barreled spaces. The relationship between completeness conditions on the underlying locally convex space and completeness conditions on its operator algebra is discussed for various operator topologies. In particular, if “large” algebras A of operators on a locally convex space SEare required to have properties necessary for a reasonable theory of A-valued functions, then these results show that A must be barreled. For more detailed discussion, we require the following familiar notions. (i) A set B C X in a locally convex space is bounded if? it is absorbed by every neighborhood N of 0: ANIB for some h > 0. (ii) Let b : r)r x (D2+ gs be a bilinear function, where the !& are locally convex. For each u E’I)~ , write b, : gz --+ 9s for the function b,(e) = b(u, v). Similarly, b,(u) = b(u, V) maps r)i into & . Then b is Zeft (respectively, right) hypocontinuous 8 every b, and b, is continu- ous, while for every bounded B, C !& , the set {b,, 1u E BL} is equi- continuous in L(g2 , g3) (respectively, for every bounded B, CT&, (b, 1e, E B,) is equicontinuous in L(z)i , 9,)).
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