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ON THE ORDER STRUCTURE OF REPRESENTABLE FUNCTIONALS

Part of: Linear spaces and algebras of operators Selfadjoint operator algebras Topological (rings and) algebras with an involution

Published online by Cambridge University Press:  04 September 2017

ZSIGMOND TARCSAY  and
TAMÁS TITKOS
ZSIGMOND TARCSAY
Affiliation:
Department of Applied Analysis, Eötvös L. University, Pázmány Péter sétány 1/c, Budapest H-1117, Hungary e-mail: tarcsay@cs.elte.hu
TAMÁS TITKOS
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13-15, Budapest H-1053, Hungary e-mail: titkos.tamas@renyi.mta.hu
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Abstract

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The main purpose of this paper is to investigate some natural problems regarding the order structure of representable functionals on *-algebras. We describe the extreme points of order intervals, and give a non-trivial sufficient condition to decide whether or not the infimum of two representable functionals exists. To this aim, we offer a suitable approach to the Lebesgue decomposition theory, which is in complete analogy with the one developed by Ando in the context of positive operators. This tight analogy allows to invoke Ando's results to characterize uniqueness of the decomposition, and solve the infimum problem over certain operator algebras.

MSC classification

Primary: 46L51: Noncommutative measure and integration
Secondary: 46K10: Representations of topological algebras with involution 47L07: Convex sets and cones of operators
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 289 - 305
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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