Constructions of Nonseparable C^∗-algebras, I: Graph CCR Algebras

Abstract

In this chapter, infinitary combinatorics is applied to study graph CCR algebras. These “twisted” reduced group C^∗-algebras associated with a Boolean group and a cocycle given by a graph are AF (approximately finite), and even AM (approximately matricial) if they are simple (i.e., if they have no proper norm-closed, two-sided, ideals). After developing structure theory, we recast results of the author and Katsura and show that in spite of their simplicity (here “simplicity” stands for “lack of complexity”), graph CCR algebras provide counterexamples to several conjectures about the structure of simple, nuclear C^∗-algebras. We construct an AM C^∗-algebras that is not UHF but it has a faithful representation on a separable Hilbert space. In every uncountable density character κ, there are 2^κ nonisomorphic graph CCR algebras with the same K-theoretic invariants as the CAR algebra. By using an independent family of subsets of ${\mathbb N}$, we construct a simple graph CCR algebra that has irreducible representations on both separable and nonseparable Hilbert spaces.