Abstract
The aim of this paper is to summarize, deepen, and comment upon some recentresults concerning restrictions and extensions of states on operator algebras. Thefirst part is focused on the question of the circumstances under which a purestate or a completely positive map restricts to a pure state on maximal Abeliansubalgebra. In the second part we present an extension theorem forStone-algebra-valued measures on quotionts of JBW algebras and discuss its consequences.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
J. F. Aarnes and R. V. Kadison, Pure states and approximate identities, Proc. Am. Math. Soc. 21 (1969) 749-752.
C. A. Akemann, Approximate units and maximal Abelian C*-subalgebras, Pacific J. Math. 33 (1970) 543-550.
W. Arveson, Subalgebras of C*-algebras, Acta Math. 123, 141-224.
B. A. Barnes, Pure states with the restriction property, Proc. Am. Math. Soc. 33 (1972) 491-494.
J. Bunce, Characters on singly generated C*-algebras, Proc. Am. Math. Soc. 25 (1970) 297-303.
D. R. Farenick and P. B. Morenz, C*-extreme points in the generalized state space of a C*-algebra, Trans. Am. Math. Soc. 349 (1997) 1725-1748.
D. R. Farenick and H. Zhou, The structure of C*-extreme points in space of completely positive linear maps on C*-algebras, Proc. Am. Math. Soc. 126 (1998) 1467-1477.
H. Hanche-Olsen and E. Stormer, Jordan Operator Algebras, Pitman, Boston (1984).
J. Hamhalter, Gleason property and extensions of states on projection logics, Bull. Lond. Math. Soc. 26, 367-372.
J. Hamhalter, Universal state space embeddability of Jordan-Banach algebras, Proc. Am. Math. Soc. 127 (1999) 131-137.
J. Hamhalter, Restricting pure states on JB algebras to maximal associative subalgebras, Tatra Mountains Math. Publ. 15 (1998) 165-171.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, New York (1986).
G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London (1979).
L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Am. Math. Soc. 68 (1950) 265-247.
P. Pta´k, On extensions of states on logics, Bull. Polish Acad. Sci. Math. 33 (1985) 493-497.
G. A. Raggio, States and composite systems in W*-algebraic quantum mechanics, Dissertation ETH, No. 6824, Zu¨rich (1981).
J. D. M. Wright, An extension theorem and a dual proof of a theorem of Gleason, J. Lond. Math. Soc. 43 (1968) 699-702.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hamhalter, J. Restricting and Extending States and Positive Maps on Operator Algebras. International Journal of Theoretical Physics 39, 721–726 (2000). https://doi.org/10.1023/A:1003654208266
Issue Date:
DOI: https://doi.org/10.1023/A:1003654208266