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

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 exists 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).