Masa’s and Certain Type I Closed Faces of C−algebras

Let A be a separable C−algebra and A its enveloping W −algebra. A result of Akemann, Anderson, and Pedersen states that if {pn} is a sequence of mutually orthogonal, minimal projections in A such that P ∞ k pn is closed, ∀k, then there is a MASA B in A such that each φn|B is pure and has a unique state extension to A, where φn is the pure state of A supported by pn. We generalize this result in two ways: We prove that B can be required to contain an approximate identity of A, and we show that the countable discrete space which underlies the result cited can be replaced by a general totally disconnected space. We consider two special kinds of type I closed faces, both related to the above, atomic closed faces and closed faces with nearly closed extreme boundary. One specific question is whether an atomic closed face always has an “isolated point”. We give a counterexample for this and also show that the answer is yes if the atomic face has nearly closed extreme boundary. We prove a complement to Glimm’s theorem on type I C∗−algebras which arises from the theory of type I closed faces. One of our examples is a type I closed face which is isomorphic to a closed face of every non-type I separable C−algebra and which is not isomorphic to a closed face of any type I C−algebra. AMS subject classification: 46L05