On the continuity of certain intertwining operators, centralizers, and positive linear functionals

Abstract

This theorem is an immediate consequence of a lemma found independently by Varopoulos [9] and Johnson [5] which generalizes Cohen's factorization theorem [1]. This lemma was used by Varopoulos to show that if A has a continuous involution, then every positive linear functional on A is continuous. Johnson used the lemma to show that every centralizer on A is continuous, and our Theorem 1 is just a slight generalization of Johnson's result. Our small contribution to this circle of ideas is to show, by mneans of a simple device, that the lemma of Varopoulos and Johnson is a corollary of a generalization of Cohen's theorem found by Hewitt [3], Curtis and Figa-Talamanca [2], and Gulick, Liu and van Rooij [4]. Furthermore, a method for simplifying the proof of Cohen's theorem has been found by Koosis [6], and, as he remarks, this method can be used equally well to simplify the proof of the theorem of Hewitt et al. In this way we obtain a quite short proof of the lemma of Varopoulos and Johnson, and so of the three continuity results mentioned above. We now state the theorem of Hewitt et al. for the case in which V is a left A-module. The statement for right modules is entirely anal-

Locations

• Proceedings of the American Mathematical Society - View - PDF