Search results for "positive linear functional"
showing 5 items of 5 documents
Extensions of hermitian linear functionals
2022
AbstractWe study, from a quite general point of view, the family of all extensions of a positive hermitian linear functional $$\omega $$ ω , defined on a dense *-subalgebra $${\mathfrak {A}}_0$$ A 0 of a topological *-algebra $${\mathfrak {A}}[\tau ]$$ A [ τ ] , with the aim of finding extensions that behave regularly. The sole constraint the extensions we are dealing with are required to satisfy is that their domain is a subspace of $$\overline{G(\omega )}$$ G ( ω ) ¯ , the closure of the graph of $$\omega $$ ω (these are the so-called slight extensions). The main results are two. The first is having characterized those elements of $${\mathfrak {A}}$$ A for which we can find a positive her…
Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals
2010
The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra \({\mathfrak{A}_0}\) of a topological *-algebra \({\mathfrak{A}}\), satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given.
Representable linear functionals on partial *-algebras
2012
A GNS-like *-representation of a partial *-algebra \({{\mathfrak A}}\) defined by certain representable linear functionals on \({{\mathfrak A}}\) is constructed. The study of the interplay with the GNS construction associated with invariant positive sesquilinear forms (ips) leads to the notions of pre-core and of singular form. It is shown that a positive sesquilinear form with pre-core always decomposes into the sum of an ips form and a singular one.
Extensions of positive linear functionals on a *-algebra
2010
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 $\sigma$.
Radon-Nikodym theorem in quasi *-algebras
2013
In this paper some properties of continuous representable linear functionals on a quasi $*$-algebra are investigated. Moreover we give properties of operators acting on a Hilbert algebra, whose role will reveal to be crucial for proving a Radon-Nikodym type theorem for positive linear functionals.