@article {
author = {Brown, Lawrence},
title = {Semicontinuity and closed faces of C*-algebras},
journal = {Advances in Operator Theory},
volume = {3},
number = {1},
pages = {17-41},
year = {2018},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.22034/aot.1611-1048},
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 $pM(A)_{{\text{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.},
keywords = {operator algebras,Semicontinuity,Closed projection,Operator convex},
url = {http://www.aot-math.org/article_43918.html},
eprint = {http://www.aot-math.org/article_43918_bf8da69fd044f09da9c3e4f4db9277c1.pdf}
}