Let G be an infinite, compact abelian group, E be a homogeneous Banach space over G and LE be the space of all continuous linear operators from E into itself equipped with the operator norm. Translation operators are isometries in E (by definition) and so the closed subalgebra mE of LE consisting of those operators which commute with all translations is well defined. It is shown that there exists a contractive projection Q of LE onto mE which is positivity preserving. Moreover, every operator QT∈mE, with T∈LE, is induced by a unique Fourier multiplier function T^∈ℓ∞Γ, where Γ is the dual group of G. In the setting of the homogeneous Banach spaces LpG, for 1≤p<∞ and G an amenable group, these results are due to W. Arendt and J. Voigt.

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 space of measures and |m|_o is given by a well known formula. Some 20 years ago L. Drewnowski and W. Wnuk asked the question (for L not Dedekind complete) if there is an m for which |m|_o exists but, |m|_o is not given by the usual formula? We show that such a measure m does indeed exist.

The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It is designed to serve as an introductory graduate level text as well as a basic reference for more established mathematicians with interests in the continually expanding areas of noncommutative analysis and probability. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the general theory of rearrangement invariant Banach lattices of measurable functions which has its roots in many areas of classical analysis related to the well-known Lp-spaces. A principal aim, therefore, is to present a general theory which contains each of these motivating areas as special cases.

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 theorem is established. Whenever T is a bounded linear operator on E satisfying M_φT=TM_φ, then also M_φ̄T=TM_φ̄.