Semicontinuity and closed faces of C*-algebras
arXiv:1312.3624
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.
arXiv commentThe 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