On the classification problem for C*-algebras

Details On the classification problem for C*-algebras On the classification problem for C*-algebras

2010-02-25

arxiv.org/abs/1002.4711v2

Mathematics

Operator Algebras

23 pages

Scientific paper

In the given article, we discuss the problem of the classification of general C$^*$-algebras. Also, it was introduced a new notions of C$^*$-algebra of von Neumann type I, C$^*$-algebras of types II and III. It is proved that any GCR-algebra is a C$^*$-algebra of von Neumann type I, and any C$^*$-algebra is a NGCR-algebra if and only if this C$^*$-algebra does not have a nonzero abelian annihilator. Also in the article there were proved that for a C$^*$-algebra $A$ there exist such unique C$^*$-subalgebras $A_I$, $A_{II}$, $A_{III}$ that $A_I$ is a C$^*$-algebra of von Neumann type I, there does not exist a nonzero abelian annihilator in the algebras $A_{II}$ and $A_{III}$, the lattice $\mathcal{P_{A_{II}}}$ of annihilators of $A_{II}$ is locally modular, the lattice $\mathcal{P_{A_{III}}}$ of annihilators of $A_{III}$ is purely nonmodular. Moreover $A_I\oplus A_{II}\oplus A_{III}$ is a C$^*$-subalgebra of $A$ and the annihilator of $A_I\oplus A_{II}\oplus A_{III}$ is the set $\{0\}$, i.e. $Ann_A(A_I\oplus A_{II}\oplus A_{III})=\{0\}$. In the final part of the article there were introduced notions of C$^*$-algebra of type I$_n$, C$^*$-algebra of types II, II$_1$, II$_\infty$ and III. Then we have proven that: any simple C$^*$-algebra of von Neumann type I is a C$^*$-algebra of type I$_n$ for some cardinal number $n$, any C$^*$-algebra of type II$_1$ is finite, any simple purely infinite C$^*$-algebra is of type III and any W$^*$-factor of type II$_\infty$ has a proper ideal $J$ such that $J$ is a simple C$^*$-algebra of type II$_\infty$. Finally it has been formulated a classification theorem for simple C$^*$-algebras.

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