Semicontinuity and closed faces of C*-algebras
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.
Keywords
Cite
@article{arxiv.1312.3624,
title = {Semicontinuity and closed faces of C*-algebras},
author = {Lawrence G. Brown},
journal= {arXiv preprint arXiv:1312.3624},
year = {2017}
}
Comments
The main change from the previous version is that 4.10 is strengthened. The original 4.10 still appears as Corollary 4.11. There are also strengthenings of 4.8 and 4.9 and a new section 5