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 J A , a largest type B closed ideal J B , a largest type C closed ideal J C as well as a largest C * -semi-finite closed ideal J sf . Among them, we have J A + J B being an essential ideal of J sf , and J A + J B + J C being an essential ideal of A. On the other hand, A/J C is always C * -semi-finite, and if A is C * -semi-finite, then A/J B is of type A. This paper is dedicated to Charles Batty on the occasion of his 60th birthday.

On the decomposition into Discrete, type II and type III $C^*$-algebras

“…In [22], we use open projections in A * * to obtain another classification scheme for C *algebras parallel to the one of Murray and von Neumann. Meanwhile, we also observe that discrete C * -algebras (as defined by Peligard and Zsidó), type II C * -algebras and type III C *algebras also form a good classification scheme, and some of the results in [22] have their counterparts in this scheme. We develop a more comprehensive theory in the current paper.…”

On the decomposition into Discrete, Type II and Type III C*-algebras

“…Moreover, most of the results in this paper are completely new, e.g., all the results in Subsections 3•3 and 3•5, as well as Sections 4 and 6 have no correspondences in [25] at all. Conversely, more than half of the results in [25] has no correspondence in the current paper neither.…”

“…We develop a more comprehensive theory in this paper. Note that the overlap materials between this paper and [25] is not significant. Actually, only the arguments of Lemma 3•2 and Theorem 5•3 have overlap with the corresponding results in [25], and some of the statements that have correspondences in [25] have different proofs here.…”

“…Actually, only the arguments of Lemma 3•2 and Theorem 5•3 have overlap with the corresponding results in [25], and some of the statements that have correspondences in [25] have different proofs here. Moreover, most of the results in this paper are completely new, e.g., all the results in Subsections 3•3 and 3•5, as well as Sections 4 and 6 have no correspondences in [25] at all. Conversely, more than half of the results in [25] has no correspondence in the current paper neither.…”

“…In [25], we use open projections in A * * to obtain another classification scheme for C *algebras parallel to the one of Murray and von Neumann. Meanwhile, we also observe that discrete C * -algebras (as defined by Peligard and Zsidó), type II C * -algebras and type III C *algebras also form a good classification scheme, and some of the results in [25] have their counterparts in this scheme. We develop a more comprehensive theory in this paper.…”