Semicontinuity and closed faces of C*-algebras
Preprint
- 11 July 2014
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.All Related Versions
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