Semicontinuity and closed faces of $C^*$-algebras

Abstract

C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785-795.] defined three concepts of semicontinuity for self-adjoint elements of $A^{**}$, the enveloping von Neumann algebra of a $C^*$-algebra $A$. We give the basic properties of the analogous concepts for elements of $pA^{**}p$, where $p$ is a closed projection in $A^{**}$. In other words, in place of affine functionals on $Q$, the quasi-state space of $A$, we consider functionals on $F(p)$, the closed face of $Q$ suppported by $p$. We prove an interpolation theorem: If $h \geq k$, where $h$ is lower semicontinuous on $F(p)$ and $k$ upper semicontinuous, then there is a continuous affine functional $x$ on $F(p)$ such that $k \leq x \leq h$. We also prove an interpolation-extension theorem: Now $h$ and $k$ are given on $Q$, $x$ is given on $F(p)$ between $h_{|F(p)}$ and $k_{|F(p)}$, and we seek to extend $x$ to $\widetilde x$ on $Q$ so that $k \leq \widetilde x \leq h$. We give a characterization of $pM(A)_{{\mathrm{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.