TY - JOUR
ID - 43918
T1 - Semicontinuity and closed faces of C*-algebras
JO - Advances in Operator Theory
JA - AOT
LA - en
SN -
A1 - Brown, Lawrence G.
Y1 - 2018
PY - 2018/01/01
VL - 3
IS - 1
SP - 17
EP - 41
KW - operator algebras
KW - Semicontinuity
KW - Closed projection
KW - Operator convex
DO - 10.22034/aot.1611-1048
N2 - 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.
UR - http://www.aot-math.org/article_43918.html
L1 - http://www.aot-math.org/pdf_43918_bf8da69fd044f09da9c3e4f4db9277c1.html
ER -