Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers

Abstract

Let M X(R) be the Banach algebra of all Fourier multipliers on a Banach function space X(R) such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X (R). For two sets Ψ, Ω ⊂ M X(R) , let Ψ Ω be the set of those c ∈ Ψ for which there exist d ∈ Ω such that the multiplier norm of χ R\[−N,N ] (c − d) tends to zero as N → ∞.In this case we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if Ω is a unital Banach subalgebra of M X(R) consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and Ψ is an arbitrary unital Banach subalgebra of M X(R) , then Ψ Ω is a also a unital Banach subalgebra of M X(R) . KeywordsFourier convolution operator • Fourier multiplier • slowly oscillating function • equivalence at infinity • Banach algebra • C * -algebra. Mathematics Subject Classification (2010) 47G10 • 42A45 • 46E30 This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/MAT/ 00297/2020 (Centro de Matemática e Aplicações) and by the SEP-CONACYT project A1-S-8793 (México).