Extensions of positive operators and functionals

Sebestyén, Zoltán; Szűcs, Zsolt; Tarcsay, Zsigmond

doi:10.1016/j.laa.2015.01.028

Cited by 9 publications

(22 citation statements)

References 12 publications

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“…Consider now a left ideal I of A and a linear functional f : I → C. In this section we provide necessary and sufficient conditions under which f admits a representable extension to A (cf. also [26] for the Banach-* algebra setting). Recall that if f : I → C is a linear functional, then we can associate an operator A : I →Ā * to f by setting (5.1)…”

Section: Functional Extensions On *-Algebrasmentioning

confidence: 99%

“…It is straightforward that π f is linear and multiplicative. To see that π f preserves involution, fix x ∈ A and a, b ∈ I , then Following the terminology of [26], we will call f N the Krein-von Neumann extension of f . Note that in the above construction we gained an explicit formula for f N and also for its values on positive elements.…”

Section: Functional Extensions On *-Algebrasmentioning

confidence: 99%

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Operators on anti‐dual pairs: Generalized Krein–von Neumann extension

Tarcsay

Titkos

2021

Mathematische Nachrichten

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The main aim of this paper is to generalize the classical concept of positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The concept of anti-duality carries an adequate structure to define positivity in a natural way, and is still general enough to cover numerous important areas where the Hilbert space theory cannot be applied. Our running example -illustrating the applicability of the general setting to spaces bearing poor geometrical features -comes from noncommutative integration theory. Namely, representable extension of linear functionals of involutive algebras will be governed by their induced operators. The main theorem, to which the vast majority of the results is built, gives a complete and constructive characterization of those operators that admit a continuous positive extension to the whole space. Various properties such as commutation, or minimality and maximality of special extensions will be studied in detail.

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Section: Functional Extensions On *-Algebrasmentioning

confidence: 99%

Section: Functional Extensions On *-Algebrasmentioning

confidence: 99%

Operators on anti‐dual pairs: Generalized Krein–von Neumann extension

Tarcsay

Titkos

2021

Mathematische Nachrichten

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“…To this aim we introduce first the concept of a positive operator from a vector space into its antidual, cf. [9]. Let A be a (not necessarily unital) * -algebra, and denote by A * andĀ * the algebraic dual and antidual of A , respectively.…”

Section: Preliminariesmentioning

confidence: 99%

Lebesgue Decomposition for Representable Functionals on *-Algebras

Tarcsay¹

2015

Glasgow Math. J.

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We present a Lebesgue-type decomposition for a representable functional on a * -algebra into absolutely continuous and singular parts with respect to an other. This generalizes the corresponding results of S. P. Gudder for unital Banach * -algebras. We also prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.2010 Mathematics Subject Classification. Primary 46L45, 47A67, Secondary 46K10.

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“…To this aim we recall briefly a modified version of the GNS construction involving the associated positive operators. For the details the reader is referred to [13,Theorem 5.3]. Consider the auxiliary Hilbert space H A obtained along the procedure of Section 4.…”

Section: Parallel Sum Of Representable Functionals On a * -Algebramentioning

confidence: 99%

On the parallel sum of positive operators, forms, and functionals

Tarcsay

2015

Acta Math. Hungar.

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The parallel sum A : B of two bounded positive linear operators A, B on a Hilbert space H is defined to be the positive operator having the quadratic form inf{(A(x − y) | x − y) + (By | y) | y ∈ H}for fixed x ∈ H. The purpose of this paper is to provide a factorization of the parallel sum of the form J A P J * A where J A is the embedding operator of an auxiliary Hilbert space associated with A and B, and P is an orthogonal projection onto a certain linear subspace of that Hilbert space. We give similar factorizations of the parallel sum of nonnegative Hermitian forms, positive operators of a complex Banach space E into its topological anti-dualĒ ′ , and of representable positive functionals on a * -algebra.1991 Mathematics Subject Classification. Primary 47B25, 47B65, Secondary 28A12, 46K10.

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Extensions of positive operators and functionals

Cited by 9 publications

References 12 publications

Operators on anti‐dual pairs: Generalized Krein–von Neumann extension

Operators on anti‐dual pairs: Generalized Krein–von Neumann extension

Lebesgue Decomposition for Representable Functionals on *-Algebras

On the parallel sum of positive operators, forms, and functionals

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