Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-21T09:35:34.591Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectrally bounded traces on C*-algebras

Published online by Cambridge University Press:  17 April 2009

Martin Mathieu
Martin Mathieu
Affiliation:
Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland e-mail: m.m.@qub.ac.uk
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.

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant M ≥ 0 such that r(T x) ≤ Mr(x) for all x ∈ E, where r (·) denotes the spectral radius. We establish the equivalence of the following properties of a unital linear mapping T from a unital C* -algebra A into its centre:

(a) T is spectrally bounded;

(b) T is a spectrally bounded trace;

(c) T is a bounded trace.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 1 , August 2003 , pp. 169 - 173
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Aupetit, B., ‘Spectrum-preserving linear mappings between Banach algebras of Jordan-Banach algebras’, J. London Math. Soc. 62 (2000), 917–924.CrossRefGoogle Scholar
[2]Aupetit, B. and Mathieu, M., ‘The continuity of Lie homomorphisms’, Studia Math. 138 (2000), 193–199.Google Scholar
[3]Brešar, M. and Mathieu, M., ‘Derivations mapping into the radical, III’, J. Funct. Anal. 133 (1995), 21–29.CrossRefGoogle Scholar
[4]Brešar, M. and Šemrl, P., ‘Linear maps preserving the spectral radius’, J. Funct. Anal. 142 (1996), 360–368.CrossRefGoogle Scholar
[5]Cui, J. and Hou, J., ‘The spectrally bounded linear maps on operator algebras’, Studia Math. 150 (2002), 261–271.CrossRefGoogle Scholar
[6]Cuntz, J. and Pedersen, G. K., ‘Equivalence and traces on C* -algebras’, J. Funct. Anal. 33 (1979), 135–164.CrossRefGoogle Scholar
[7]Curto, R.E. and Mathieu, M., ‘Spectrally bounded generalized inner derivations’, Proc. Amer. Math. Soc. 123 (1995), 2431–2434.CrossRefGoogle Scholar
[8]Dixmier, J., Von Neumann algebras (North-Holland, Amsterdam, 1969).Google Scholar
[9]Mathieu, M., ‘Spectrally bounded operators on simple C* -algebras’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[10]Mathieu, M. and Schick, G.J., ‘First results on spectrally bounded operators’, Studia Math. 152 (2002), 187–199.CrossRefGoogle Scholar
[11]Mathieu, M. and Schick, G.J., ‘Spectrally bounded operators from von Neumann algebras’, J. Operator Theory 49 (2003) (to appear).Google Scholar
[12]Pop, C., ‘Finite sums of commutators’, Proc. Amer. Math. Soc. 130 (2002), 3039–3041.CrossRefGoogle Scholar
[13]Šemrl, P., ‘Spectrally bounded linear maps on B(H)’, Quart. J. Math Oxford 49 (1998), 87–92.CrossRefGoogle Scholar