A Murray–von Neumann Type Classification of C*-algebras

Abstract

We define type A, type B, type C as well as C*-semi-finite C*-algebras.

It is shown that a von Neumann algebra is a type A, type B, type C or C*-semi-finite C*-algebra if and only if it is, respectively, a type I, type II, type III or semi-finite von Neumann algebra. Any type I C*-algebra is of type A (actually, type A coincides with the discreteness as defined by Peligrad and Zsidó), and any type II C*-algebra (as defined by Cuntz and Pedersen) is of type B. Moreover, any type C C*-algebra is of type III (in the sense of Cuntz and Pedersen). Conversely, any separable purely infinite C*-algebra (in the sense of Kirchberg and Rørdam) with either real rank-zero or stable rank-one is of type C.

We also prove that type A, type B, type C and C*-semi-finiteness are stable under taking hereditary C*-subalgebras, multiplier algebras and strong Morita equivalence. Furthermore, any C*-algebra A contains a largest type A closed ideal JA, a largest type B closed ideal JB, a largest type C closed ideal JC as well as a largest C*-semi-finite closed ideal Jsf. Among them, we have JA + JB being an essential ideal of Jsf, and JA + JB + JC being an essential ideal of A. On the other hand, A/JC is always C*-semi-finite, and if A is C*-semi-finite, then A/JB is of type A.

This paper is dedicated to Charles Batty on the occasion of his 60th birthday

Mathematics Subject Classification (2010). 46L05, 46L35.