Semicontinuity and Closed Faces of C∗-algebras

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 ≥ 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 ≤ x ≤ 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 x̃ on Q so that k ≤ x̃ ≤ h. We give a characterization of pM(A)sap in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity. 1. Definitions, notations, and basic properties For a C∗-algebra A, S = S(A) denotes the state space of A and Q = Q(A) the quasi–state space, Q(A) = {φ ∈ A∗ : φ ≥ 0 and ‖φ‖ ≤ 1}. E. Effros [13] showed that norm closed faces of Q(A) containing 0 correspond one–to-one to projections p in A∗∗, via F (p) = {φ ∈ Q : φ(1− p) = 0}. Then p is called closed if F (p) is weak∗ closed and open if p is the support projection of a hereditary Copyright 2016 by the Tusi Mathematical Research Group. Date: Received: Nov. 1, 2016; Accepted: Mar. 4, 2017. 2010 Mathematics Subject Classification. Primary 46L05.