Some Simple $C^*$-Algebras Constructed As Crossed Products with Discrete Outer Automorphism Groups

Abstract

An analogue for C*-algebras is given of the theorem of von Neumann (Theorem VIII of [24]) that the crossed product of a commutative von Neumann algebra by a discrete group acting freely and ergodically is a factor.The method of proof also works for certain noncommutative C*-algebras, and so in these cases one obtains also an analogue of Kallman's noncommutative generalization of von Neumann's theorem (3.3 of [18]).§ 1. Introduction It seems to be a reasonable conjecture that the C*-algebra crossed product, suitably defined, of a simple C*-algebra with unit by a group of outer automorphisms should be simple.For a finite group (and an arbitrary simple ring with unit) this was proved by Azumaya in [2].It follows from Azumaya's work that for any group the algebraic crossed product is simple (see also 1.1 of [16]).If the group is not amenable, then the largest pre-C*-algebra norm on the algebraic crossed product does not in general yield a simple C*-algebra (see 5.2 of [25]) 9 but in [25] Zeller-Meier introduced a smaller norm, determining what he called the reduced crossed product, and showed that if the automorphisms satisfy a condition which in general is rather more than being outer (and if the C*-algebra is separable and the group of automorphisms countable) then the reduced crossed product is simple.Recently, Cuntz in [6], and Bratteli in [3] gave examples of simple crossed products of simple C*-algebras by groups of outer automorphisms not satisfying