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

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

2010-09-14

arxiv.org/abs/1009.2592v1

Mathematics

Operator Algebras

14 pages, to appear in JFA

Scientific paper

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 $[0,1]/\{0,1\}$.

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