A Murray-von Neumann type classification of $C^*$-algebras

Details A Murray-von Neumann type classification of $C^*$-algebras A Murray-von Neumann type classification of $C^*$-algebras

2011-12-07

arxiv.org/abs/1112.1455v1

Mathematics

Operator Algebras

29 pages. Any comment is welcome

Scientific paper

We define type $\ta$, type $\tb$, type $\tc$ as well as $C^*$-semi-finite $C^*$-algebras. It is shown that a von Neumann algebra is a type $\ta$, type $\tb$, type $\tc$ 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 $\ta$ (actually, type $\ta$ coincides with the discreteness as defined by Peligrad and Zsid\'{o}), and any type II $C^*$-algebra (as defined by Cuntz and Pedersen) is of type $\tb$. Moreover, any type $\tc$ $C^*$-algebra is of type III (in the sense of Cuntz and Pedersen). Conversely, any purely infinite $C^*$-algebra (in the sense of Kirchberg and R{\o}rdam) with real rank zero is of type $\tc$, and any separable purely infinite $C^*$-algebra with stable rank one is also of type $\tc$. We also prove that type $\ta$, type $\tb$, type $\tc$ 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 $\ta$ closed ideal $J_\ta$, a largest type $\tb$ closed ideal $J_\tb$, a largest type $\tc$ closed ideal $J_\tc$ as well as a largest $C^*$-semi-finite closed ideal $J_\SF$. Among them, we have $J_\ta + J_\tb$ being an essential ideal of $J_\SF$, and $J_\ta + J_\tb + J_\tc$ being an essential ideal of $A$. On the other hand, $A/J_\tc$ is always $C^*$-semi-finite, and if $A$ is $C^*$-semi-finite, then $A/J_\tb$ is of type $\ta$. Finally, we show that these results hold if type $\ta$, type $\tb$, type $\tc$ and $C^*$-semi-finiteness are replaced by discreteness, type II, type III and semi-finiteness (as defined by Cuntz and Pedersen), respectively.

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