Positivity 11 (2007), 417–432
2007 Birkh¨auser Verlag Basel/Switzerland
Positive Functionals on some Spaces
Abstract. The aim of this work is to study norm preserving extensions of
positive functionals on some spaces of fractions. The main result is stated in
the case of unitary complex Banach algebras with involution. Moreover, we
deal with C*-algebras and the commutative case as well. An application is
Mathematics Subject Classiﬁcation (2000). Primary 46E99; Secondary 46M40.
Keywords. positive functionals, spaces of fractions, *-algebras, C*-algebras,
representative measure, extending positive forms, moment problem.
1. Introduction and preliminaries
The spaces of fractions was recently used by F.H.Vasilescu in the context of mo-
ment problems in . He extended positive functionals on some spaces of fractions
constructed from C(Ω), when Ω is a compact Hausdorﬀ topological space.
In this work we study the possibility of generalizing his result to a non com-
mutative context, which is the aim of the second section. In the third one we reﬁne
the hypotheses in the C*-algebra case. Then we present some related results, in
the commutative case, in the forth section. In the last one we give an application
to moment problems.
Throughout the paper, all the spaces dealt with are supposed to be in the
category of normed vector spaces either over the real or over the complex ﬁeld.
Now we give the general context of our work, then we recall the construction
of the space of fraction, after that, we state the problem which is the matter of
our discussion in this paper.
Let E be an ordered vector space. The set of positive elements forms a cone
that will be denoted by C. A linear functional ω on E is called positive if it is
positive on C.
We denote by L(E) the algebra of all bounded linear operators on E, endowed
with the usual norm. An operator A ∈ L(E) is said to be positive if A(b) ≥ 0for
all b ∈ E,b ≥ 0.