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 classes of extensions (positive, positively regular, absolutely convergent) are constructed. The obtained results are applied to the commutative integration theory to recover from the abstract setup the well-known extensions of Lebesgue integral and, in noncommutative integration theory, for introducing a generalized non absolutely convergent integral of operators measurable w. r. to a given trace $\sigma$.

Bongiorno, B., Trapani, C., Triolo, S. (2010). Extensions of positive linear functionals on a *-algebra. ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 40(6), 1745-1777 [10.1216/RMJ-2010-40-6-1745].

### 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 classes of extensions (positive, positively regular, absolutely convergent) are constructed. The obtained results are applied to the commutative integration theory to recover from the abstract setup the well-known extensions of Lebesgue integral and, in noncommutative integration theory, for introducing a generalized non absolutely convergent integral of operators measurable w. r. to a given trace $\sigma$.
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Bongiorno, B., Trapani, C., Triolo, S. (2010). Extensions of positive linear functionals on a *-algebra. ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, 40(6), 1745-1777 [10.1216/RMJ-2010-40-6-1745].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/40142