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.

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.