Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T03:41:19.852Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Rank Operators and Functional Calculus on Hilbert Modules over Abelian C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Dan Kucerovsky
Dan Kucerovsky*
Affiliation:
Université Paris 6 Laboratoire de Mathematiques Fondamentales aile 46–00, (URA 747) 4, pl. Jussieu F7525 Paris Cedex 5 France, e-mail: kucerov@mathp6.jussieu.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the problem: If K is a compact normal operator on a Hilbert module E, and f ∈ C0(SpK) is a function which is zero in a neighbourhood of the origin, is f(K) of finite rank? We show that this is the case if the underlying C*-algebra is abelian, and that the range of f(K) is contained in a finitely generated projective submodule of E.

Keywords

Primary: 55R5047A6047B38
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 2 , 01 June 1997 , pp. 193 - 197
Copyright
Copyright © Canadian Mathematical Society 1997

References

[APT] Akemann, C., Pedersen, G. and Tomiyama, J., Multipliers of CŁ-algebras, J. Funct. Anal. 13 (1973).Google Scholar
[BT] Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Math. 82, Springer-Verlag, 1982.Google Scholar
[Dix] Dixmier, J. and Douady, A., Champs continus d’espaces hilbertiens et de C Ł-algèbre, Bull. Soc. Math. France 91 (1963), 227–284.Google Scholar
[Dup] Dupré, M. J., Hilbert Bundles with Infinite Dimensional Fibres, Mem. Amer. Math. Soc. 148 (1974), 165–176.Google Scholar
[Fra] Frank, M., Self-duality and CŁ-reflexivity of Hilbert CŁ-moduli, Z. Anal. Anwendungen 9 (1990), 165–176.Google Scholar
[GK] Gokhberg, I. and Krein, I. M., Introduction to nonselfadjoint operators, Trans. Amer. Math. Soc. 18, Providence, New York, 1969.Google Scholar
[Kas1] Kasparov, G. G., The operator K-functor and extension of CŁ–algebras, Math. USSR-Izv. 16 (1981), 513–636.Google Scholar
[Kas2] Kasparov, G. G., Hilbert CŁ-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), 133–150.Google Scholar
[Kuc] Kucerovsky, D., Kasparov Products in KK-theory, and unbounded operators, with applications to index theory, Magdalen College, University of Oxford (thesis), 1994.Google Scholar
[Man] Manuilov, V. M., Diagonalization of compact opertors in Hilbert modules over WŁ-algebras of finite type, Uspekhi Mat. Nauk 49 (1994), 159–160.Google Scholar
[MP] Mingo, J. A. and Phillips, W. J., Equivariant triviality theorems for Hilbert CŁ-modules, Proc.Amer.Math. Soc. 91 (1984), 225–230.Google Scholar
[Pas] Paschke, W., Inner Product modules over BŁ-algebras, Trans.Amer.Math. 182 (1973), 443–468.Google Scholar
[Skan] Skandalis, G., Kasparov's bivariant K-theory and applications, Exposition. Math. 9 (1991), 193–250.Google Scholar
[Wegg] Wegge-Olsen, N. E., K-Theory and CŁ-Algebras, Oxford Univ. Press, Oxford, 1993.Google Scholar