A simple C*-algebra with a finite and an infinite projection

Abstract

An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite. Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras. Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-algebra arises as a crossed product of an inductive limit of type I C*-algebras by an action of the integers.

Revised version (contains a nuclear example), revised again August 2002

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• Classification of tiling $C^{\ast }$-algebras
• Nuclear dimension of graph C${}^{\ast }$-algebras with Condition~(K)
• A classification of finite simple amenable $\mathcal{Z}$-stable $C^{\ast }$-algebras, I: $C^{\ast }$-algebras with generalized tracial rank one
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