Lefschetz Numbers and Geometry of Operators in W*-modules

TL;DR

This paper extends the concept of Lefschetz numbers to a broader setting involving W*-algebras and more general endomorphisms, providing new theoretical tools in operator algebra geometry.

Contribution

It generalizes Lefschetz number definitions to W*-modules without requiring group representations, introducing larger target groups and new operator theory results.

Findings

Defined Lefschetz numbers in larger groups as $K_0(A)\otimes\C$.

Developed new results on Hilbert W*- and C*-modules.

Provided insights into bounded module operators on these modules.

Abstract

The main goal of the present paper is to generalize the results of~\cite{TroLNM,TroBoch} in the following way: To be able to define $K_0(A)\o\C$-valued Lefschetz numbers of the first type of an endomorphism $V$ on a C*-elliptic complex one usually assumes that $V=T_g$ for some representation $T_g$ of a compact group $G$ on the C*-elliptic complex. We try to refuse this restriction in the present paper. The price to pay for this is twofold: (i) $ $ We have to define Lefschetz numbers valued in some larger group as $K_0(A)\o\C$. (ii) We have to deal with W*-algebras instead of general unital C*-algebras. To obtain these results we have got a number of by-product facts on the theory of Hilbert W*- and C*-modules and on bounded module operators on them which are of independent interest.