%0 Journal Article
%T Semicontinuity and closed faces of C*-algebras
%J Advances in Operator Theory
%I Tusi Mathematical Research Group (TMRG)
%Z 2538-225X
%A Brown, Lawrence G.
%D 2018
%\ 01/01/2018
%V 3
%N 1
%P 17-41
%! Semicontinuity and closed faces of C*-algebras
%K operator algebras
%K Semicontinuity
%K Closed projection
%K Operator convex
%R 10.22034/aot.1611-1048
%X 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 $hgeq k$, 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 $kleq xleq h$. 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 $kleqwidetilde xleq h$. We give a characterization of $pM(A)_{{text{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.
%U http://www.aot-math.org/article_43918_bf8da69fd044f09da9c3e4f4db9277c1.pdf