Extensions of hermitian linear functionals

Abstract

We 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 hermitian slight extension of $$\omega $$ ω , giving the range of the possible values that the extension may assume on these elements; the second one is proving the existence of maximal positive hermitian slight extensions. We show as it is possible to apply these results in several contexts: Riemann integral, Infinite sums, and Dirac Delta.

Maximal extensions of a linear functional

“…In this paper, after showing that maximal extensions of linear functionals are necessarily everywhere defined, we revisit widely positive, fully positive and absolute convergent extensions already discussed in [2] and prove several new features that emerge from the discussion. Applications to extensions of Riemann integral on continuous functions are also examined.…”

“…In this paper, we continue the analysis, undertaken in [2], [3] of the possibility of extending a positive hermitian linear functional ω, defined on a dense *-subalgebra A 0 of a topological *-algebra (in general, without unit), with topology τ and continuous involution * , to some elements of A. Moreover, we resume the notion of positive regular slight extension that closely reminds the construction of the Lebesgue integral or Segal's construction of noncommutative integration [16].…”

“…To construct hermitian extensions (see [2,3]), we define H ω as the collection of all subspaces H ∈ S ω for which the following additional condition holds (h3) (a, ℓ) ∈ H if and only if (a * , ℓ) ∈ H, and then we proceed like in the case of the construction of slight extensions. In [2], it is proved that all maximal hermitian extensions share the same domain. More precisely: Proposition 2.4.…”

“…Thus, in an abstract set-up it makes sense to consider extensions enjoying appropriate properties. As in [2] and [3], the starting point is the notion of slight extension, which is treated for general linear maps in Köthe's book [6]. As application of the developed ideas, we report interesting results concerning infinite sums (see [2]).…”

“…The situation however changes deeply if one looks for positive extensions. The case of fully positive and widely positive extensions considered in [2] is revisited from this point of view. Examples mostly taken from the theory of integration are discussed.…”