Extensions of positive linear functionals on a *-algebra

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Abstract

The family of all extensions of a nonclosable hermitian positive linear functional defined on a dense *-subalgebra $\Ao$ of a topological *-algebra $\A[\tau]$is studied with the aim of finding extensions that behave regularly. Under suitable assumptions, special classesof extensions (positive, positively regular, absolutely convergent) are constructed. The obtained results areapplied to the commutative integration theory to recover from the abstract setup the well-known extensions ofLebesgue integral and, in noncommutative integration theory, for introducing a generalized non absolutelyconvergent integral of operators measurable w. r. to a given trace $\sigma$.
Lingua originaleEnglish
pagine (da-a)1745-1777
Numero di pagine33
RivistaDefault journal
Volume40
Stato di pubblicazionePublished - 2010

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Linear Functionals
Algebra
Topological Algebra
Linear Functional
Subalgebra
Trace
Operator

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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title = "Extensions of positive linear functionals on a *-algebra",
abstract = "The family of all extensions of a nonclosable hermitian positive linear functional defined on a dense *-subalgebra $\Ao$ of a topological *-algebra $\A[\tau]$is studied with the aim of finding extensions that behave regularly. Under suitable assumptions, special classesof extensions (positive, positively regular, absolutely convergent) are constructed. The obtained results areapplied to the commutative integration theory to recover from the abstract setup the well-known extensions ofLebesgue integral and, in noncommutative integration theory, for introducing a generalized non absolutelyconvergent integral of operators measurable w. r. to a given trace $\sigma$.",
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AU - Trapani, Camillo

AU - Bongiorno, Benedetto

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AB - The family of all extensions of a nonclosable hermitian positive linear functional defined on a dense *-subalgebra $\Ao$ of a topological *-algebra $\A[\tau]$is studied with the aim of finding extensions that behave regularly. Under suitable assumptions, special classesof extensions (positive, positively regular, absolutely convergent) are constructed. The obtained results areapplied to the commutative integration theory to recover from the abstract setup the well-known extensions ofLebesgue integral and, in noncommutative integration theory, for introducing a generalized non absolutelyconvergent integral of operators measurable w. r. to a given trace $\sigma$.

UR - http://hdl.handle.net/10447/40142

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