Extensions of positive linear functionals on a topological *-algebra
Let \(\omega\) be a given hermitian positive linear functional defined on a dense subalgebra \(\mathcal{U}_0\) of a topological \(*\)-algebra \(\mathcal{U}\). This paper is mainly concerned with the problem of extending \(\omega\) to some subspaces of \(\mathcal{U}\) in the case when \(\omega\) is nonclosable, and is specialized in the slight extension which was given for arbitrary linear maps in [\textit{G. Köthe}, ``Topological vector spaces II'' (Grundlehren der mathematischen Wissenschaften 237; New York, Heidelberg, Berlin: Springer-Verlag) (1979; Zbl 0417.46001)]. Unlike in the case when \(\omega\) is closable, the authors show that there exist infinitely many positive extensions of \(\omega\) whose graph is contained in the closure of the graph of \(\omega\). They also characterize the maximal ones and prove that, if \(\omega\) admits an absolutely convergent positively regular extension \(\breve{\omega}\) which is maximal positive, then this extension is unique. In the last two sections, they give applications to commutative integration and noncommutative integration. They show that the Henstock-Kurzweil integral is one of the extensions of the Riemann integral defined in an abstract way, and among all possible extensions of the integral, the Lebesgue one is maximally absolutely convergent. In the application to noncommutative integration, the authors explicitly construct one of the extensions of the normal semifinite faithful trace \(\sigma\) defined on a von Neumann algebra beyond the space \(L^1(\sigma)\) of measurable operators which behaves in a similar fashion to the Henstock-Kurzweil integral in the commutative case. The notion of measurable operators was given by \textit{I. E. Segal} [Ann. Math. 57, 401--457 (1953; Zbl 0051.34201)].
scientific article; zbMATH DE number 5837624
scientific article