Abstract
We give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory. These consist of positive parts of unit balls for C*-algebras and their dual spaces, and for von Neumann algebras and their pre-duals; of self-adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes. All these faces are shown to be semi-exposed and naturally paired with a polar face in the dual (or pre-dual) space
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