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 $\sigma$.
Original languageEnglish
Pages (from-to)1745-1777
Number of pages33
JournalRocky Mountain Journal of Mathematics
Volume40
Publication statusPublished - 2010

All Science Journal Classification (ASJC) codes

  • General Mathematics

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