C. R. Math. Rep. Acad. Sci. Canada Vol. 32 (4) 2010, pp. 106–119
December 30, 2010
Dan Kučerovský, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3; email: dan@math.unb.ca
P.W. Ng, Department of Mathematics, University of Louisiana at Lafayette, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, LA, 70504–1010 USA; email: png@louisiana.edu
Abstract/Résumé:
Let \(\mathcal{A}\) be a unital, separable, simple \(C^*\)-algebra. Denote by \(G := U \bigl( \mathcal{M}(\mathcal{A} \otimes \mathcal{K}) \bigr)\) the unitary group of the multiplier algebra of \(\mathcal{A} \otimes \mathcal{K}\), given the strict topology. Then the following conditions are equivalent:
(1) \(\mathcal{A}\) is a nuclear \(C^*\)-algebra.
(2) \(G\) is an amenable topological group.
(3) \(G\) is an extremely amenable topological group.
(4) The Kasparov extension of \(\mathcal{A} \otimes \mathcal{K}\) is absorbing.
(5) The Lin and Kasparov extensions of \(\mathcal{A} \otimes \mathcal{K}\) are approximately unitarily equivalent (with unitaries coming from \(\mathcal{M}( \mathcal{A} \otimes \mathcal{K})\)).
(6) The Kasparov extension of \(S \mathcal{A} \otimes \mathcal{K}\) is absorbing.
(7) The suspended Lin extension and the Kasparov extension, of \(S \mathcal{A} \otimes \mathcal{K}\), are approximately unitarily equivalent (with unitaries coming from \(\mathcal{M}(S \mathcal{A} \otimes \mathcal{K})\)).
(8) Every purely large extension of \(\mathcal{A} \otimes \mathcal{K}\) is absorbing.
(9) Every properly purely large extension of \(\mathcal{A} \otimes \mathcal{K}\) is absorbing.
Soit \(\mathcal{A}\) une \(C^*\)-algèbre unifère, séparable et simple et soit \(\mathcal{M}(\mathcal{A} \otimes \mathcal{K})\) l’algèbre des multiplicateurs de \(\mathcal{A} \otimes \mathcal{K}\). Dénotons par \(G\) le groupe unitaire de \(\mathcal{M}(\mathcal{A} \otimes \mathcal{K})\) muni de la topologie stricte. Nous démontrons plusieurs caractérisations équivalentes de la nucléarité de \(\mathcal{A}\). En particulier, nous prouvons l’équivalence des conditions suivantes:
(1) \(\mathcal{A}\) est une \(C^*\)-algèbre nucléaire.
(2) \(G\) est un groupe topologique moyennable.
(3) L’extension de Kasparov de \(\mathcal{A} \otimes \mathcal{K}\) est absorbante.
AMS Subject Classification: General theory of $C^*$-algebras 46L05
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