Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Pure state extensions and compressibility of the $ l\sb 1$-algebra


Author: Betül Tanbay
Journal: Proc. Amer. Math. Soc. 113 (1991), 707-713
MSC: Primary 46L30; Secondary 47C15, 47D25
MathSciNet review: 1062394
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L30, 47C15, 47D25

Retrieve articles in all journals with MSC: 46L30, 47C15, 47D25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1062394-0
PII: S 0002-9939(1991)1062394-0
Keywords: Pure state, pure state extension, maximal abelian subalgebra, compression, $ {l_1}$-algebra, ultrafilter, $ \beta ({\mathbf{N}})$
Article copyright: © Copyright 1991 American Mathematical Society