Some Directed Subsets of C*-algebras and Semicontinuity Theory

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 $a\le h-\delta \b 1$ 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 $h=\b 1$, is that if $a$ and $b$ are self-adjoint in $A$, $a\le h$, and $b\le h-\delta \b 1$ for $\delta >0$, then there is a self-adjoint $c$ in $A$ such that $c\le h, a\le c$, and $b\le c$.