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E-theory for C[0, 1]-algebras with finitely many singular points

Published online by Cambridge University Press:  03 March 2014

Marius Dadarlat  and
Prahlad Vaidyanathan
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Abstract

We study the E-theory group E[0, 1](A, B) for a class of C*-algebras over the unit interval with finitely many singular points, called elementary C[0, 1]-algebras. We use results on E-theory over non-Hausdorff spaces to describe E[0, 1](A, B) where A is a sky-scraper algebra. Then we compute E[0, 1](A, B) for two elementary C[0, 1]-algebras in the case where the fibers A(x) and B(y) of A and B are such that E1 (A(x), B(y)) = 0 for all x, y ∈ [0, 1]. This result applies whenever the fibers satisfy the UCT, their K0-groups are free of finite rank and their K1-groups are zero. In that case we show that E[0, 1](A, B) is isomorphic to Hom(0(A), 0(B)), the group of morphisms of the K-theory sheaves of A and B. As an application, we give a streamlined partially new proof of a classification result due to the first author and Elliott.

Keywords

Continuous fields of C*algebrasE-theoryClassification of C*algebrasPrimary 46L35Secondary 19K3546L8046M20
Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 2 , April 2014 , pp. 249 - 274
Copyright
Copyright © ISOPP 2014 

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References

1.Bauval, Anne, R K K(X)-nucléarité (d'après G. Skandalis). K-Theory 13(1) (1998), 23–40.CrossRefGoogle Scholar
2.Dadarlat, M., Fiberwise K K-equivalence of continuous fields of C*-algebras. J. K-Theory 3(2) (2009), 205–219.Google Scholar
3.Dadarlat, M. and Elliott, G. A., One-parameter continuous fields of Kirchberg algebras. Comm. Math. Phys. 274(3) (2007), 795–819.Google Scholar
4.Dadarlat, M. and Meyer, R., E-theory for C*-algebras over topological spaces. J. Funct. Anal. 263(1) (2012), 216–247.Google Scholar
5.Dadarlat, M. and Pasnicu, C., Continuous fields of Kirchberg C*-algebras. J. Funct. Anal. 226(2) (2005), 429–451.Google Scholar
6.Dixmier, J., C* algebras 15. North Holland Publishing Company, 1977.Google Scholar
7.Kirchberg, E., Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren. In C*-algebras (Münster, 1999), pages 92–141. Springer, Berlin, 2000.Google Scholar
8.Park, E. and Trout, J.. Representable E-theory for C 0(X)-algebras. J. Funct. Anal. 177(1) (2000), 178–202.CrossRefGoogle Scholar
9.Rørdam, M. and Størmer, E.. Classification of nuclear C*-algebras. Entropy in operator algebras, Encyclopaedia of Mathematical Sciences 126. Springer-Verlag, Berlin, 2002. Operator Algebras and Non-commutative Geometry 7.Google Scholar
10.Rosenberg, J. and Schochet, C.. The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor. Duke Math. J. 55(2) (1987), 431–474.CrossRefGoogle Scholar