A Murray-von Neumann Type Classification of C∗-algebras

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. Moreover, 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), any purely infinite C∗-algebra (in the sense of Kirchberg and Rørdam) with real rank zero is of type C, and any separable purely infinite C∗-algebra with stable rank one is also of type C. We also prove that type A, type B, type C and C∗-semi-finiteness are stable under 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. Finally, we show that these results hold if type A, type B, type C and C∗-semifiniteness are replaced by discreteness, type II, type III and semi-finiteness (as defined by Cuntz and Pedersen), respectively.