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

2006/11/14 by Marius Dădărlat, Dadarlat, Marius · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.math/0611408

openalex publication_date 2006/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 KK_X(A,B) is invertible if and only if all its fiberwise components σ_x∈ 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 O₂. 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.

Citations

• On the KK-theory of strongly self-absorbing C*-algebras

Cited by

• Reduced C^*-algebras of Product Systems -- an E₀-semigroup and a Groupoid perspective

Related

• Continuous C*-algebras over topological spaces
• Continuous Fields of Kirchberg C*-algebras
• Continuous fields of C*-algebras over finite dimensional spaces
• Continuous fields of properly infinite C*-algebras
• Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O₂