Orthogonal complements and endomorphisms of Hilbert modules and C*-elliptic complexes

Summary

Introduction

In the present paper we discuss some properties of endomorphisms of C*- Hilbert modules and C*-elliptic complexes. The main results of this paper can be considered as an attempt to answer the question: what kinds of good properties can one expect for an operator on a Hilbert module, which represents an element of a compact group? These results are new, but we have to recall some first steps made by us before to make the present paper self-contained.

In §2 we define the Lefschetz numbers “of the first type” of C*-elliptic complexes, taking values in K₀(A) ⊗ℂ, A being a complex C*-algebra with unity, and prove some properties of them.

The averaging theorem 3.2 was discussed in brief in and was used there for constructing an index theory for C*-elliptic operators. In this theorem we do not restrict the operators to admit a conjugate, but after averaging they even become unitary. This raises the following question: is the condition on an operator on a Hilbert module to represent an element of a compact group so strong that it automatically has to admit a conjugate?

The example in section 4 gives a negative answer to this question. Also we get an example of closed submodule in Hilbert module which has a complement but has no orthogonal complement.

In §5 we define the Lefschetz numbers of the second type with values in HC₀(A). We prove that these numbers are connected via the Chern character in algebraic K-theory. These results were discussed in and we only recall them.