Semicontinuity and Multipliers of C*-Algebras

Abstract

In [5] C. Akemann and G. Pedersen defined four concepts of semicontinuity for elements of A**, the enveloping W*-algebra of a C*-algebra A. For three of these the associated classes of lower semicontinuous elements are , and (notation explained in Section 2), and we will call these the classes of strongly lsc, middle lsc, and weakly lsc elements, respectively. There are three corresponding concepts of continuity: The strongly continuous elements are the elements of A itself, the middle continuous elements are the multipliers of A, and the weakly continuous elements are the quasi-multipliers of A. It is natural to ask the following questions, each of which is three-fold. (Q1) Is every lsc element the limit of a monotone increasing net of continuous elements? (Q2) Is every positive lsc element the limit of an increasing net of positive continuous elements? (Q3) If h ≧ k, where h is lsc and k is usc, does there exist a continuous x such that h ≧ x ≧ k?