Lefschetz Numbers and Geometry of Operators in W*-modules
Michael Frank, Evgenij V. Troitsky
TL;DR
The paper tackles the problem of defining $K_0(A)\otimes{\bf C}$-valued Lefschetz numbers for endomorphisms on $A$-elliptic complexes without requiring a group-representation origin for the endomorphism. It develops the Hilbert W*-module framework to define and analyze two Lefschetz-number types, $L_1$ in $K_0(A)_S$ and $L_{2l}$ in $HC_{2l}(A)$, and establishes a Chern-character bridge between them. It also investigates obstructions in the general C*-case that prevent a straightforward generalization, and explains how Paschke–Lin machinery and $A^{**}$-extensions become necessary to realize the theory in a broader setting. The work yields structural results for Hilbert W*-modules, $A$-Fredholm theory, and new Lefschetz-type invariants valued in $K_0(A)$ and cyclic homology, with potential implications for noncommutative geometry and operator-algebraic index theory.
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 $K_0(A)ø\C$-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 $K_0(A)ø\C$. (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.