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continuous fields

Journal of K-Theory (1)

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Fiberwise KK-equivalence of continuous fields of C*-algebras
Marius Dadarlat
Journal:

Journal of K-Theory / Volume 3 / Issue 2 / April 2009

Published online by Cambridge University Press:

28 May 2008, pp. 205-219

Print publication:

April 2009
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Let A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.