The modulus of a vector measure

It is known that if L is a Dedekind complete Riesz space and (Ω, Σ) is a measurable space, then the partially ordered linear space of all L-valued, finitely additive and order bounded vector measures m on Σ is also a Dedekind complete Riesz space (for the natural operations). In particular, the modulus |m|_o of m exists in this...

A Fuglede type theorem for Fourier multiplier operators

Let E be a translation invariant Banach function space over an infinite compact abelian group G and M_φ be a Fourier multiplier operator (with symbol φ) acting on E. It is assumed that E has order continuous norm and that E is reflection invariant (which ensures that φ̄ is also a multiplier symbol for E). The following Fuglede type...

Free and projective Banach lattices

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice...

Mean ergodic operators and reflexive Fréchet lattices

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it...

Properties (u) and (V*) of Pelczynski in symmetric spaces of ? -measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of ? -measurable operators are respectively equivalent to properties (u) and (V?) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These...

Completely positive compact operators on non-commutative symmetric spaces

Under natural conditions, it is shown that a completely positive operator between two non-commutative symmetric spaces of ? -measurable operators which is dominated in the sense of complete positivity by a completely positive compact operator is itself compact.

R -bounded Representations of L 1 ( G )

Products of commuting Boolean algebras of projections and Banach space geometry

Characterizations of conditional expectation-type operators

As is well-known, conditional expectation operators on various function spaces exhibit a number of remarkable properties related either to the underlying order structure of the given function space, or to the metric structure when the function space is equipped with a norm. Such operators are necessarily positive projections which are averaging...

Algebras of unbounded scalar-type spectral operators.

If P: ? -* J?f (X) is a closed spectral measure in the quasicomplete locally convex space X and if T is a densely defined linear operator in X with domain invariant under each operator of the form /? fdP, with / a complex bounded ?-measurable function then T is closable and there exists a complex ?-measurable function / such that the closure of...

The space of extended orthomorphisms in a Riesz space

We study the space Orth°°(L) of extended orthomorphisms in an Archimedean Riesz space L and its analogies with the complete ring of quotients of a commutative ring with unit element. It is shown that for any uniformly complete /-algebra A with unit element, Orth°°(?) is isomorphic with the complete ring of quotients of A. Furthermore, it is...