Pure state extensions and compressibility of the $l_ 1$-algebra
HTML articles powered by AMS MathViewer
- by BetΓΌl Tanbay PDF
- Proc. Amer. Math. Soc. 113 (1991), 707-713 Request permission
Abstract:
In 1958 Kadison and Singer proved that not every pure state on a continuous maximal abelian subalgebra (masa) of the algebra $\mathcal {B}(\mathcal {H})$ of all bounded linear operators on a separable complex Hilbert space $\mathcal {H}$, has a unique pure state extension to $\mathcal {B}(\mathcal {H})$ [5]. They conjectured the same result is true for discrete masas, and although the question remains open, it was shown by Anderson in 1978 to be equivalent to determining whether all operators in $\mathcal {B}(\mathcal {H})$ are compressible. We define in this paper the ${l_1}$-subalgebra $\mathcal {M}$ of $\mathcal {B}(\mathcal {H})$, and show that all operators in $\mathcal {M}$ are compressible. Hence every pure state on a discrete masa has a unique pure state extension to $\mathcal {M}$.References
- Joel Anderson, Extensions, restrictions, and representations of states on $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 249 (1979), no.Β 2, 303β329. MR 525675, DOI 10.1090/S0002-9947-1979-0525675-1
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185 F. Drake, Set theory, Elsevier, North-Holland, 1974. H. Halpern, V. Kaftal, and G. Weiss, Matrix pavings in $\mathcal {B}(\mathcal {H})$, Operator Theory: Adv. Appl., vol. 24, 1987.
- Richard V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383β400. MR 123922, DOI 10.2307/2372748
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630 B. Tanbay, Extensions of pure states on algebras of operators, Ph.D. Thesis, University of California, Berkeley, 1989.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 707-713
- MSC: Primary 46L30; Secondary 47C15, 47D25
- DOI: https://doi.org/10.1090/S0002-9939-1991-1062394-0
- MathSciNet review: 1062394