Some directed subsets of C*-algebras and semicontinuity theory

Abstract

The main result concerns a sigma-unital C*-algebra A, a strongly lower semicontinuous element h of A**, the enveloping von Neumann algebra, and the set of self-adjoint elements a of A such that a \le h - delta 1 for some delta > 0, where 1 is the identity of A**. The theorem is that this set is directed upward. It follows that if this set is non-empty, then h is the limit of an increasing net of self-adjoint elements of A. A complement to the main result, which may be new even if h = 1, is that if a and b are self-adjoint in A, a \le h, and b \le h - delta 1 for delta > 0, then there is a self-adjoint c in A such that c \le h, a \le c, and b \le c.