MASA's and certain type I closed faces of C*-algebras

TL;DR

This paper generalizes a result on pure states and MASAs in C*-algebras, explores properties of type I closed faces, and provides counterexamples and structural theorems related to these faces.

Contribution

It extends previous theorems by including approximate identities and totally disconnected spaces, and analyzes properties of atomic and nearly closed extreme boundary faces.

Findings

A new generalization of pure states and MASAs with approximate identities

Counterexample showing atomic closed faces may lack isolated points

A complement to Glimm's theorem on type I C*-algebras

Abstract

A result of Akemann, Anderson, and Pedersen states that if a sequence of pure states of a C*-algebra A approaches infinity in a certain sense, then there is a MASA B such that each of the states has the unique extension property with respect to B. We generalize this in two ways: We prove that B can be required to contain an approximate identity of A, and we show that the discrete space which underlies the result cited can be replaced with a 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 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.