Proper asymptotic unitary equivalence in $\KK$-theory and projection lifting from the corona algebra

TL;DR

This paper extends the concept of essential codimension within $ ext{KK}$-theory to establish conditions for unitary equivalence of projections via asymptotic unitaries, with applications to corona algebras.

Contribution

It generalizes essential codimension using $ ext{KK}$-theory and proves unitary equivalence of projections through proper asymptotic unitaries, applying results to corona algebras.

Findings

Unitary of the form 'identity + compact' can realize projection equivalence when essential codimension vanishes.

The generalized notion of essential codimension applies to various $C^*$-algebras and corona algebra projections.

Results facilitate projection lifting in corona algebras of $C(X) ensor B$.

Abstract

In this paper we generalize the notion of essential codimension of Brown, Douglas, and Fillmore using $\KK$-theory and prove a result which asserts that there is a unitary of the form `identity + compact' which gives the unitary equivalence of two projections if the `essential codimension' of two projections vanishes for certain $C\sp*$-algebras employing the proper asymptotic unitary equivalence of $\KK$-theory found by M. Dadarlat and S. Eilers. We also apply our result to study the projections in the corona algebra of $C(X)\otimes B$ where $X$ is $[0,1]$, $(-\infty, \infty)$, $[0,\infty)$, and $[0,1]/\{0,1\}$.