Fourier Multipliers in Homogeneous Banach Spaces

Let G be an infinite, compact abelian group, E be a homogeneous Banach space over G and L( E) L E 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 m( E) m E of L( E) L E consisting of those operators which commute with all translations is well defined. It is shown that there exists a contractive projection Q Q of L( E) L E onto m( E) m E which is positivity preserving. Moreover, every operator Q( T) ∈ m( E) Q T ∈ m E , with T∈ L( E) T ∈ L E , is induced by a unique Fourier multiplier function T̂∈ ∞( Γ ) T ^ ∈ ℓ ∞ Γ , where Γ Γ is the dual group of G . In the setting of the homogeneous Banach spaces Lᵖ( G) L p G , for 1≤ p<∞ 1 ≤ p < ∞ and G an amenable group, these results are due to W. Arendt and J. Voigt.