Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-08T04:28:57.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of states of C* -algebras, II

Published online by Cambridge University Press:  14 November 2011

R. J. Archbold ,
J. W. Bunce  and
K. D. Gregson
R. J. Archbold
Affiliation:
Department of Mathematics, University of Aberdeen, The Edward Wright Building, Dunbar Street, Aberdeen AB9 2TY
J. W. Bunce
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, U.S.A.
K. D. Gregson
Affiliation:
Department of Mathematics, University of Aberdeen, The Edward Wright Building, Dunbar Street, Aberdeen AB9 2TY
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let A be a unital C*-algebra and let B be an abelian C*-subalgebra containing the identity of A. For any pure state h of B let Fh be the set of states of A which restrict to h on B. Necessary and sufficient conditions are given for an element x in A to have the property that, for each h, x is unable to distinguish between distinct elements of Fh. By specializing, this leads to a new proof of a theorem giving necessary and sufficient conditions for Fh to be a singleton for each h.

It is also shown that if A is postliminal and π(B) is a maximal abelian C*-subalgebra of π(B) for each irreducible representation π of A then Fh is a Choquet simplex for each h.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 92 , Issue 1-2 , 1982 , pp. 113 - 122
Copyright
Copyright © Royal Society of Edinburgh 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alfsen, E. M.. Compact convex sets and boundary integrals (Berlin: Springer, 1971).CrossRefGoogle Scholar
2Akemann, C. A.. Approximate units and maximal abelian C*-algebras. Pacific J. Math. 33 (1970), 543–550.CrossRefGoogle Scholar
3Anderson, J.. A maximal abelian subalgebra of the Calkin algebra with the extension property. Math. Scand. 42 (1978), 101–110.Google Scholar
4Anderson, J.. Extreme points in sets of positive linear maps on B(H). J. Functional Analysis 31 (1979), 195–217.Google Scholar
5Anderson, J.. Extensions, restrictions, and representations of states on C*-algebras. Trans. Amer. Math. Soc. 249 (1979), 303–329.Google Scholar
6Anderson, J.. A conjecture concerning the pure states of B(H) and a related theorem. In Proceedings of the Vth International conference on operator theory, Timisoara and Herculane, Romania, to appear.Google Scholar
7Archbold, R. J.. Extensions of states of C*-algebras. J. London Math. Soc. 21 (1980), 351–354.Google Scholar
8Batty, C. J. K.. Abelian faces of state spaces of C*-algebras. Comm. Math. Phys. 75 (1980), 43–50.CrossRefGoogle Scholar
9Bratelli, O..and Robinson, D. W.. Operator algebras and quantum statistical mechanics, I (Berlin: Springer, 1979).Google Scholar
10Bunce, J. W.. Finite operators and amenable C*-algebras. Proc. Amer. Math. Soc. 56 (1976), 145–151.Google Scholar
11Cuntz, J.. Automorphisms of certain simple C*-algebras, preprint.Google Scholar
12Dang-Ngoc, N.. On the integral representation of states on a C*-algebra. Comm. Math. Phys. 40 (1975), 223–233.Google Scholar
13Dixmier, J.. Les C*-algèbres et lews représentations, 2nd edn (Paris: Gauthier-Villars, 1969).Google Scholar
14Dixmier, J.. Les Algèbres d'opérateurs dans Vespace Hilbertien, 2nd edn (Paris: Gauthier-Villars, 1969).Google Scholar
15Dunford, N.. and Schwartz, J. T.. Linear Operators, Part I (New York: Interscience, 1966).Google Scholar
16Effros, E. G..and Hahn, F.. Locally compact transformation groups and C*-algebras. Mem. Amer. Math. Soc. 75 (1967).Google Scholar
17Gregson, K. D.. Dissertation, University of Aberdeen, in preparation.Google Scholar
18Henrichs, R. W.. Maximale Integralzerlegungen invarianter positiv definiter Funktionen auf diskreten Gruppen. Math. Ann. 208 (1974), 15–31.Google Scholar
19Kadison, R. V..and Singer, I. M.. Extensions of pure states. Amer. J. Math. 81 (1959), 547–564.CrossRefGoogle Scholar
20Krauss, F.. and Lawson, T. C.. Examples of homogeneous C*-algebras. Mem. Amer. Math. Soc. 148 (1974).Google Scholar
21Lanford, O.. and Ruelle, D.. Integral representation of invariant states on B*-algebras. J. Math. Phys. 8 (1967), 1460–1463.Google Scholar
22Pedersen, G. K.. C*-algebras and their automorphism groups (London: Academic Press, 1979).Google Scholar
23Power, S. C.. Simplicity of C*-algebras of minimal dynamical systems. J. London Math. Soc. 18 (1978), 534–538.CrossRefGoogle Scholar
24Reid, G. A.. On the Calkin representations. Proc. London Math. Soc. 23 (1971), 547–564.Google Scholar
25Takesaki, M.. Theory of operator algebras I (Berlin: Springer, 1979).Google Scholar