关于C* 代数的分类问题
On the classification problem for C*-algebras
摘要：
在给定的文章中，它是引入新的概念的C $^*$ -代数的冯诺依曼型I和C $^*$ -代数的类型I $_n$，II，II $_1$，II $_\infty$和III。证明了任何GCR-代数都是von Neumann I型C $^*$ -代数，C $^*$ -代数是NGCR-代数当且仅当该C $^*$ -代数不具有非零Abel零化子.也证明了一个类似的定理分解的冯诺依曼代数的子代数的类型I，II和III。最后证明了：对于某个基数$n$，每个VonNeumann I型CPDMATHABMM-因子是I型$_n$的CPDMATHABMM-代数，每个II型$_1$的单$^*$ -代数是有限的，每个纯无限$^*$ -代数是III型的，每个II型$^*$ -因子有一个II型$_\infty$的单CPDMATHABMM-子代数.最后给出了C $^*$ -因子的分类定理。
英文摘要：
In the given article it is introduced new notions of a C$^*$-algebra of von Neumann type I and C$^*$-algebras of types I$_n$, II, II$_1$, II$_\infty$ and III. It is proved that any GCR-algebra is a C$^*$-algebra of von Neumann type I, and a C$^*$-algebra is an NGCR-algebra if and only if this C$^*$-algebra does not have a nonzero Abelian annihilator. Also an analog of the theorem on decomposition of a von Neumann algebra to subalgebras of types I, II and III is proved. In the final part it is proved that every C$^*$-factor of von Neumann type I is a C$^*$-algebra of type I$_n$ for some cardinal number $n$, every simple C$^*$-algebra of type II$_1$ is finite, every simple purely infinite C$^*$-algebra is of type III and every W$^*$-factor of type II$_\infty$ has a simple C$^*$-subalgebra of type II$_\infty$. Finally it is formulated a classification theorem for C$^*$-factors.
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