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