Constructions of Nonseparable C^∗-algebras, II

Abstract

Other examples of nonseparable C^∗-algebras are constructed in this chapter. We start with Akemann’s C^∗-algebra with no abelian approximate unit. This is followed by a strengthening of a result of Akemann and Weaver due to the author and Hirshberg. It gives a counterexample to Glimm’s dichotomy for nonseparable C^∗-algebras: For every n ≤ℵ₀ there exists a simple C^∗-algebra of density character ℵ₁ with exactly n unitarily inequivalent irreducible representations. The case n = 1 is a counterexample to Naimark’s problem. These results use Jensen’s $\diamondsuit _{\aleph _1}$, and it is not known whether they can be proved in ZFC. The chapter concludes with a study of ${\mathrm {C}^{*}_{r}}(F_{\kappa })$, the reduced group algebra of the free group with κ generators. For every κ, this C^∗-algebra has only separable abelian C^∗-subalgebras (Popa), and every two pure states are conjugate by an automorphism (Akemann–Wassermann–Weaver). Both results apply to ${\mathrm {C}^{*}_{r}}$(Γ), where Γ is the free product of any family of nontrivial countable groups.