Hilbert modules over a $C^*$-algebra of stable rank one

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (2) 2007, pp. 48–51

June 30, 2007

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that for countably generated Hilbert C$^*$-modules over a C$^*$-algebra of stable rank one (i.e., a C$^*$-algebra in which the invertible elements are dense) the relation of compact inclusion up to isomorphism is cancellative, in a certain weak but natural sense. This generalizes the well-known fact that cancellation is valid in the abelian semigroup of isomorphism classes of finitely generated projective modules over such a C$^*$-algebra.

Il est démontré que la relation d’inclusion compacte entre modules de Hilbert dénombrablement engendrés sur une C$^*$-algèbre de rang stable égal à un est cancellative, dans un sens faible mais naturel. Ceci généralise un résultat bien connu pour le cas des modules projectifs finiment engendrés.

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

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