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March 25, 2015 By

Quasitraces on Exact C*-algebras are Traces

C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (2-3) 2014, pp. 67–92

June 30, 2014

Uffe Haagerup, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark; e-mail: haagerup@math.ku.dk

Abstract/Résumé:

It is shown that all 2-quasitraces on a unital exact \(C^*\)-algebra are traces. As consequences one gets: (1) Every stably finite exact unital \(C^*\)-algebra has a tracial state, and (2) if an \(AW^*\)-factor of type \(II_1\) is generated (as an \(AW^*\)-algebra) by an exact \(C^*\)-subalgebra, then it is a von Neumann \(II_1\)-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and Rørdam to prove that \(RR(A)=0\) for every simple non-commutative torus of any dimension.

On démontre que toute 2-quasitrace sur une C*-algèbre exacte à élément unité est une trace. On en déduit les deux conséquences suivantes: (1) Toute C*-algèbre stablement finie et exacte possède un état tracial, et (2) si un AW*-facteur de type \(II_1\) est engendré (comme AW*-algèbre) par une sous-C*-algèbre exacte, il est une algèbre de von Neumann. Ceci est une solution partielle à un problème bien connu de Kaplansky. Le résultat principal a été utilisé par Blackadar, Kumjian, et Rørdam pour démontrer que \(RR(A) = 0\) pour tout tore non-commutatif simple de dimension quelconque.

Keywords: Quasitraces, classification of C*-algebras, exact C*-algebras
AMS Subject Classification: General theory of $C^*$-algebras 46L05

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