Split exactness, operator homotopy and stable uniqueness in KK

TL;DR

This paper explores fundamental properties of the KK-functor, providing simplified proofs for the Kasparov product, its associativity, and the stable uniqueness theorem, enhancing understanding of operator homotopy and split exactness.

Contribution

It introduces new short proofs for key KK-theory properties and connects operator homotopy with homotopy using quasihomomorphisms, simplifying existing arguments.

Findings

Short proof of Kasparov product existence

Proof of associativity of the Kasparov product

Derivation of the stable uniqueness theorem from operator homotopy

Abstract

We develop important properties of the KK-functor on the basis of split exactness. In particular we discuss two slightly different short proofs for the existence of the Kasparov product and its associativity. We use the approach with quasihomomorphisms to obtain a short proof of the fact that operator homotopy implies homotopy . Using an idea of Gabe-Szabo we also deduce from this the 'stable uniqueness theorem' of Dadarlat-Eilers by a very short argument.