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ON THE ORDER STRUCTURE OF REPRESENTABLE FUNCTIONALS

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

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: [email protected]
TAMÁS TITKOS
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13-15, Budapest H-1053, Hungary e-mail: [email protected]
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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

REFERENCES

1. Anderson, W. N. Jr. and Trapp, G. E., The extreme points of a set of positive semidefinite operators, Linear Algebra Appl. 106 (1988), 209–217.CrossRefGoogle Scholar
2. Ando, T., Lebesgue-type decomposition of positive operators, Acta Sci. Math. (Szeged) 38 (3–4) (1976), 253–260.Google Scholar
3. Ando, T., Problem of infimum in the positive cone, in Analytic and geometric inequalities and applications, (Rassias, T. M. and Srivastava, H. M., Editors), Math. Appl., vol. 478 (Kluwer Acad. Publ., Dordrecht, 1999), 1–12.CrossRefGoogle Scholar
4. Eriksson, S. L. and Leutwiler, H., A potential theoretic approach to parallel addition, Math. Ann. 274 (2) (1986), 301–317.Google Scholar
5. Green, W. L. and Morley, T. D., The extreme points of order intervals of positive operators, Adv. Appl. Math. 15 (3) (1994), 360–370.Google Scholar
6. Gudder, S., A Radon-Nikodym theorem for *-algebras, Pacific J. Math. 80 (1) (1979), 141–149.CrossRefGoogle Scholar
7. Gudder, S. and Moreland, T., Infima of Hilbert space effects, Linear Algebra Appl. 286 (1–3) (1999), 1–17.Google Scholar
8. Hassi, S., Sebestyén, Z. and de Snoo, H., Lebesgue type decompositions for nonnegative forms, J. Funct. Anal. 257 (12) (2009), 3858–3894.Google Scholar
9. Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras I. (Academic Press, New York, 1983).Google Scholar
10. Kosaki, H., Lebesgue decomposition of states on a von Neumann algebra, Am. J. Math. 107 (3) (1985), 697–735.Google Scholar
11. Palmer, T. W., Banach algebras and the general theory of *-algebras II (Cambridge University Press, Cambridge, 2001).Google Scholar
12. Pekarev, E. L., Shorts of operators and some extremal problems, Acta Sci. Math. (Szeged) 56 (1–2) (1992), 147–163.Google Scholar
13. Riesz, F., Sur quelques notions fondamentales dans la théorie générale des opérations linéaires, Ann. of Math. 41 (1) (1940), 174–206.Google Scholar
14. Sakai, S., C*-Algebras and W*-algebras (Springer-Verlag, Berlin-Heidelberg, New York, 1971).Google Scholar
15. Sebestyén, Z., On representability of linear functionals on *-algebras, Period. Math. Hungar. 15 (3) (1984), 233–239.Google Scholar
16. Szűcs, Zs., On the Lebesgue decomposition of positive linear functionals, Proc. Am. Math. Soc. 141 (2) (2013), 619–623.Google Scholar
17. Tarcsay, Zs., Lebesgue decomposition for representable functionals on *-algebras, Glasgow Math. J. 58 (2) (2016), 491–501.CrossRefGoogle Scholar
18. Tarcsay, Zs., On the parallel sum of positive operators, forms, and functionals, Acta Math. Hungar. 147 (2) (2015), 408–426.Google Scholar
19. Titkos, T., Ando's theorem for nonnegative forms, Positivity 16 (4) (2012), 619–626.Google Scholar