B. de Pagter

Symmetric Functionals and Singular Traces

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.

Averaging operators and positive contractive projections

Abstract The main topic of the present paper is a generalization of two theorems about positive contractive projections on C 0 ( X ) (due to Seever and Kelley, resp.) to the more general structure of ƒ-algebras. The theorems in question are the following: if A is an Archimedean semiprime ƒ-algebra with the Stone condition, then every positive contractive projection T in A satisfies the identity T ( a · Tb ) = T ( Ta · Tb ) for all a , b ϵ A (Seever); furthermore, T is averaging if and only if the range of T is a subalgebra (Kelley). Both theorems hold in particular for an Archimedean unital ƒ-algebra. In contrast to the original proofs of Seever and Kelley in the C 0 (X) -case, our proofs are purely algebraic and order theoretical.

The order bidual of lattice ordered algebras

Abstract The main topic of the present paper is a systematic investigation of the second order dual A ″ of an Archimedean ƒ-algebra A with point separating order dual A ′. It is shown that in the case that A has a unit element, the equality A ″ = ( A ′)′ n holds, where ( A ′)′ n is the collection of all order continuous linear functionals on A ′. It turns out that in general ( A ′)′ n , equipped with the Arens multiplication is an ƒ-algebra again. Necessary and sufficient conditions are derived for ( A ′)′ n to be semiprime and for ( A ′)′ n to have a unit element with respect to this multiplication.

The non-commutative Yosida-Hewitt decomposition revisited

In this paper, a new approach to the non-commutative YosidaHewitt decomposition is presented in the general setting of non-commutative symmetric spaces of τ -measurable operators affiliated with semi-finite von Neumann algebras. The principal theorem permits the systematic study of the spaces of normal and singular functionals in this general setting. These results are used to study the properties of elements of order continuous norm and of absolutely continuous norm.

Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

Let M be a Bade complete (or cr-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is M equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with a(T) c K) in X? It is shown that if X is separable, then M always coincides with such a resolution of the identity. For certain restrictions on M some positive results are established in non-separable spaces X. An example is given for which M is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

Lipschitz continuity of the absolute value in preduals of semifinite factors

We prove a weak-type estimate for the absolute value mapping in the preduals of semifinite factors which extends an earlier result of Kosaki for the trace class.

Operator Theory in Function Spaces and Banach Lattices

This volume is dedicated to A.C. Zaanen, one of the pioneers of functional analysis, and eminent expert in modern integration theory and the theory of vector lattices, on the occasion of his 80th birthday. The book opens with biographical notes, including Zaanens curriculum vitae and list of publications. It contains a selection of original research papers which cover a broad spectrum of topics about operators and semigroups of operators on Banach lattices, analysis in function spaces and integration theory. Special attention is paid to the spectral theory of operators on Banach lattices; in particular, to the one of positive operators. Classes of integral operators arising in systems theory, optimization and best approximation problems, and evolution equations are also discussed. The book will appeal to a wide range of readers engaged in pure and applied mathematics.

Maharam extensions of positive operators and f-modules

The principal result of this paper is the construction of simultaneous extensions of collections of positive linear operators between vector lattices to interval preserving operators (i.e., Maharam operators). This construction is based on some properties of so-called f-modules. The properties and structure of these extension spaces is discussed in some detail.

Unconditional Decompositions and Schur-type Multipliers

The purpose of this paper is to present a general framework for the treatment of Schur-type multipliers. Several applications are given, in particular, multiplier results for non-commutative L p -spaces are obtained, and we indicate a general approach to the theory of double operator integrals.

Vilenkin systems and generalized triangular truncation operator

We study actions of the Vilenkin group ∏∞k=0ℤm(k) onLp-spaces associated with a semi-finite von Neumann algebra М, via a generalized triangular truncation operator. The systems of eigenspaces that arise contain the classical unbounded Vilenkin systems and we show that such systems with the inverse lexicographic enumeration form Schauder decompositions in all reflexive non-commutativeLp-spaces. This is a non-commutative analogue of a theorem of Paley for unbounded Vilenkin systems, which in the classical setting is due to W.S. Young, F. Schipp and P. Simon.