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Winter 2018 Semicontinuity and closed faces of $C^*$-algebras
Lawrence G. Brown
Adv. Oper. Theory 3(1): 17-41 (Winter 2018). 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 $pM(A)_{{\mathrm{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.

Citation

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Lawrence G. Brown. "Semicontinuity and closed faces of $C^*$-algebras." Adv. Oper. Theory 3 (1) 17 - 41, Winter 2018. https://doi.org/10.22034/aot.1611-1048

Information

Received: 1 November 2016; Accepted: 4 March 2017; Published: Winter 2018
First available in Project Euclid: 5 December 2017

zbMATH: 1385.46036
MathSciNet: MR3730337
Digital Object Identifier: 10.22034/aot.1611-1048

Subjects:
Primary: 46L05

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 1 • Winter 2018
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