Development of index theory for C-algebras

设M是第II_1-型因子，N是M.V.Jones的子因子，Jones利用耦合常数引入Jones指标[M：N]作为M的N-模。Kosaki和Longo将琼斯指数理论推广到III子因素。作者将Jones指标理论推广到简单C^&lt；**&gt；-代数上，引入了Hilbert C^&lt；**&gt；-双模的概念，用Pimsner-Popa型不等式代替了Rieffel非本原双模中的结合条件。Hilbert C^&lt；**&gt；-双模的指数度量了非本原双模的失效。我们还从C^&lt；**&gt；-代数的等变Morita等价的动机和子因子的orbilold构造出发，研究了可数离散群在Hilbert C^&lt；**&gt；-双模上的作用。我们考虑Hilbert C^&lt；**&gt；-双模的交叉积，证明了某种自由的遍历作用保证了交叉积的不可约性。我们还存在Hilbert C^&lt；**&gt；-双模的Takesaki-Takai对偶。给出了A型和D型动态图在双模水平上的对偶性的一个具体例子。研究了Hilbert C^&lt；**&gt；-双模及其共轭的迭代张量积的不可约分解的融合规则。

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.