Development of index theory for C-algebras

设M为II_1型因子，N为M. V. Jones的子因子，利用耦合常数引入Jones指数[M：N]作为M作为N-模的一定秩. Kosaki和Longo将Jones指数理论扩展到III型子因子。研究者将Jones指标理论推广到了简单C^**>-代数上，在本研究中我们引入了Hilbert C^**>-双模的概念，用Pimsner-Popa型不等式代替了Rieffel的内插双模中的结合性条件。Hilbert C^**-双模的指数度量了不可导双模的失效。从C^**-代数的等变Morita等价和子因子的orbifold构造出发，研究了可数离散群在Hilbert C^**-双模上的作用.我们考虑了Hilbert C^**-双模的交叉积，并证明了某种自由遍历作用保证了交叉积的不可约性。对于Hilbert C^<**>-双模，我们也有Takesaki-Takai对偶。给出了双偶层次上A型Dynkin图和D型Dynkin图的对偶性的一个具体例子。研究了Hilbert C^**>-双模及其共轭的迭代张量积不可约分解的一个融合规则。

Let M be a type II_1-factor and N a subfactor of M.V.Jones introduced Jones index [M : N] as a certain rank of M as N-module using the coupling constant. Kosaki and Longo extended Jones index theory for type III subfactors. The head investigator had extended Jones index theory for simple C^<**>-lgebras.In this research we introduced the notion of Hilbert C^<**>-bimodule by replacing the associativity condition in Rieffel's imprimitivity bimodules by the Pimsner-Popa type ineqality. The index of Hilbert C^<**>-bimodules measures the failure from imprimitivity bimodules. We also studied actions of countable discrete groups on Hilbert C^<**>-bimodules from the motivation of equivariant Morita equivalence for C^<**>-algebras and orbifold construction for subfactors. We considered crossed products of Hilbert C^<**>-bimodules and showed that a certain kind of free and ergodic action guarantees the irreducibility of the crossed product. We also have Takesaki-Takai duality for Hilbert C^<**>-bimodules. We gave a concrete example of the duality of Dynkin diagrams of type A and type D in the level of bimoules. We investigated a fusion rule of irreducible decomposition of iterative tensor products of a Hilbert C^<**>-bimodule and its conjugate.