L2-index、KK理论和联系
L2-index, KK-theory, and connections
摘要：
设$M$是紧致流形. $D$是$M$上的Dirac型微分算子.设$A$是一个$C^*$ -代数.给定$M$上的$A$ -模块的束$W$（具有连接），运算符$D$可以与该束绞合。然后可以使用$A$上的迹来定义这个扭曲算子的数值指数。我们证明了这个指数的一个明确的公式。我们的结果补充了Mishchenko-Fomenko指标定理在相同的情况下有效。如果迹只定义在稠密的全纯闭子代数$\mathcal{B}$上，我们建立了这些显式指数公式的推广。作为推论，我们证明了一个推广的Atiyah $L^2$ -指数定理，如果扭丛是平坦的.实际上有许多不同的方法来定义这些数字指数。从它们的结构来看，根本不清楚它们是否一致。本文的一个重要部分是对这一说法的完整证明。特别是，我们建立（众所周知，但没有很好地记录）平等的Atiyah的定义的$L^2$指数与K-理论定义。在$A$是2型冯诺依曼代数的情况下，我们特别强调了中心值指标的计算和解释。这完全包含了关于扭曲算子的指数的所有K-理论信息。我们的一些计算是在双变KK理论的框架下进行的。
英文摘要：
Let $M$ be a compact manifold. and $D$ a Dirac type differential operator on $M$. Let $A$ be a $C^*$-algebra. Given a bundle $W$ of $A$-modules over $M$ (with connection), the operator $D$ can be twisted with this bundle. One can then use a trace on $A$ to define numerical indices of this twisted operator. We prove an explicit formula for this index. Our result does complement the Mishchenko-Fomenko index theorem valid in the same situation. We establish generalizations of these explicit index formulas if the trace is only defined on a dense and holomorphically closed subalgebra $\mathcal{B}$. As a corollary, we prove a generalized Atiyah $L^2$-index theorem if the twisting bundle is flat. There are actually many different ways to define these numerical indices. From their construction, it is not clear at all that they coincide. An important part of the paper are complete proofs of this statement. In particular, we establish the (well known but not well documented) equality of Atiyah's definition of the $L^2$-index with a K-theoretic definition. In case $A$ is a von Neumann algebra of type 2, we put special emphasis on the calculation and interpretation of the center valued index. This completely contains all the K-theoretic information about the index of the twisted operator. Some of our calculations are done in the framework of bivariant KK-theory.
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