Hostname: page-component-89b8bd64d-rbxfs Total loading time: 0 Render date: 2026-05-08T05:31:13.313Z Has data issue: false hasContentIssue false
  • English
  • Français

LEBESGUE DECOMPOSITION FOR REPRESENTABLE FUNCTIONALS ON *-ALGEBRAS

Part of: General theory of linear operators Selfadjoint operator algebras Topological (rings and) algebras with an involution

Published online by Cambridge University Press:  21 July 2015

ZSIGMOND TARCSAY
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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

MSC classification

Primary: 46L45: Decomposition theory for $C^*$-algebras
Secondary: 47A67: Representation theory 46K10: Representations of topological algebras with involution

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 491 - 501
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Darst, R. B., A decomposition of finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31–37.Google Scholar
2.Dixmier, J., C*-algebras, Translated from the French by Jellett, F., North-Holland Mathematical Library, vol. 15 (North-Holland publishing Co., Amsterdam–New York–Oxford, 1977).Google Scholar
3.Gudder, S. P., A Radon-Nikodym theorem for *-algebras, Pacific J. Math. 80 (1979), 141–149.Google Scholar
4.Halmos, P. R., Measure theory (Springer-Verlag, New York, 1974).Google Scholar
5.Hassi, S., Sebestyén, Z., de Snoo, H. S. V. and Szafraniec, F. H., A canonical decomposition for linear operators and linear relations, Acta Math. Hungar. 115 (2007), 281–307.CrossRefGoogle Scholar
6.Hassi, S., Sebestyén, Z. and de Snoo, H., Lebesgue type decompositions for nonnegative forms, J. Funct. Anal. 257 (2009), 3858–3894.CrossRefGoogle Scholar
7.Kosaki, H., Remarks on Lebesgue-type decomposition of positive operators, J. Operator Theory 11 (1984), 137–143.Google Scholar
8.Palmer, T. W., Banach algebras and the general theory of *-algebras II, (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
9.La Russa, C. and Triolo, S., Radon–Nikodym theorem in quasi *-algebras, J. Operator Theory 69 (2013), 423–433.CrossRefGoogle Scholar
10.Sebestyén, Z., On representability of linear functionals on *-algebras, Period. Math. Hungar. 15 (1984), 233–239.Google Scholar
11.Sebestyén, Z., Szűcs, Zs. and Tarcsay, Zs., Extensions of positive operators and functionals, Linear Algebra Appl. 472 (2015), 54–80.CrossRefGoogle Scholar
12.Sebestyén, Z., Tarcsay, Zs. and Titkos, T., Lebesgue decomposition theorems, Acta Sci. Math. (Szeged) 79 (2013), 219–233.CrossRefGoogle Scholar
13.Szűcs, Zs., The singularity of positive linear functionals, Acta Math. Hungar. 136 (2012), 138–155.CrossRefGoogle Scholar
14.Szűcs, Zs., On the Lebesgue decomposition of positive linear functionals, Proc. Amer. Math. Soc. 141 (2013), 619–623.Google Scholar
15.Szűcs, Zs., The Lebesgue decomposition of representable forms over algebras, J. Operator Theory 70 (2013), 3–31.CrossRefGoogle Scholar
16.Tarcsay, Zs., Lebesgue-type decomposition of positive operators, Positivity 17 (2013), 803–817.Google Scholar
17.Tarcsay, Zs., A functional analytic proof of the Lebesgue-Darst decomposition theorem, Real Anal. Exchange 39 (2013), 219–226.CrossRefGoogle Scholar
18.Titkos, T., Lebesgue decomposition of contents via nonnegative forms, Acta Math. Hungar. 140 (2013), 151–161.CrossRefGoogle Scholar