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Journal of Operator Theory

Volume 70, Issue 1, Summer 2013  pp. 3-31.

The Lebesgue decomposition of representable forms over algebras

Authors Zsolt Szucs
Author institution: Department of Applied Analysis, Eotvos L. University, Pazmany Peter setany 1/C, 1117 Budapest, Hungary

Summary:  This paper contains new results on Lebesgue type decompositions of nonnegative forms on complex algebras. We introduce the concept of representable forms, and we discuss the representability of the regular and the singular parts. A result on topologically irreducible representations in the context of the Lebesgue decomposition is included. We prove a general Lebesgue decomposition theorem for representable positive functionals on $*$-algebras by the above-mentioned results on representable forms. This theory was studied by other authors in very different ways. We also clarify the correspondence of this kind of decompositions. The Lebesgue decomposition theorem for measures follows from our results. A completion of the proof of a theorem due to S. Hassi, Z. Sebesty\'en and H. de Snoo is included.

Keywords:  forms, parallel sums, Lebesgue type decompositions, representations, irreducible representations, positive functionals, positive measures

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