Semicontinuity and closed faces of C*-algebras

TL;DR

This paper explores semicontinuity concepts for elements of the double dual of C*-algebras, providing new theorems on interpolation, extension, and characterizations related to operator convexity.

Contribution

It introduces analogous semicontinuity concepts for elements supported by closed projections and establishes new interpolation and characterization theorems.

Findings

Proves an interpolation theorem for semicontinuous functions on closed faces.

Provides an extension theorem for semicontinuous functions from faces to the whole space.

Characterizes certain subalgebras and convexity properties using semicontinuity.

Abstract

C. Akemann and G. Pedersen 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 supported 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 x is between h and k. 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 x on Q so that x is between h and k. We give a characterization of p(M(A)_sa)p in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.