Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T05:08:04.786Z Has data issue: false hasContentIssue false
  • English
  • Français

The Structure of C*-Convex Sets

Published online by Cambridge University Press:  20 November 2018

Phillip B. Morenz
Phillip B. Morenz*
Affiliation:
Department of Pure Mathematics, University of Waterloo Waterloo, Ontario N2L 3G1
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.

Compact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

Keywords

47A1215A3052A01
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 1007 - 1026
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Arveson, W. B., Subalgebras of C* - algebras II, Acta Math. 128(1972), 271–308.Google Scholar
2. Choi, M.-D., Completely positive linear maps on complex matrices, Linear Algebra Appl. 10(1975), 285–290.Google Scholar
3. Farenick, D. R., Krein-Milman type problems for compact matricially convex sets, Linear Algebra Appl. 162–164(1992), 325–334.Google Scholar
4. Farenick, D. R., C* - convexity and matricial ranges, Canad. J. Math. 44(1992), 280–297.Google Scholar
5. Farenick, D. R., Matricial extensions of the numerical range: a brief survey, Linear and Multilinear Algebra, 34(1993), 197–211.Google Scholar
6. Farenick, D. R. and Morenz, P. B., C* - extreme points of some compact C * - convex sets, Proc. Amer. Math. Soc. 118(1993), 765–776.Google Scholar
7. Hoppenwasser, A., Moore, R. L. and Paulsen, V. I., C* - extreme points, Trans. Amer. Math. Soc. 266(1981), 291–307.Google Scholar
8. Loebl, R. I. and Paulsen, V. I., Some remarks on C* - convexity, Linear Algebra Appl. 35(1981), 63–78.Google Scholar
9. Morenz, P. B., The structure of C* - convex sets, Dissertation, University of Toronto, 1992.Google Scholar
10. Salinas, N., Extensions of C* - algebras and n-normal operators, Bull. Amer. Math. Soc. 82(1976), 143–46.Google Scholar
11. Smith, R. R. and Ward, J. D., Matrix ranges for Hilbert space operators, Amer. J. Math. 102(1980), 1031–1081.Google Scholar