Fiberwise KK-equivalence of continuous fields of C*-algebras

Details Fiberwise KK-equivalence of continuous fields of C*-algebras Fiberwise KK-equivalence of continuous fields of C*-algebras

2006-11-14

arxiv.org/abs/math/0611408v2

Mathematics

Operator Algebras

13 pages

Scientific paper

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 $\sigma$ of the parametrized Kasparov group KK_X(A,B) is invertible if and only if all its fiberwise components $\sigma_x\in KK(A(x),B(x))$ 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 O_2. 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.

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