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

Let $$\mathcal {M}_{X(\mathbb {R})}$$ be the Banach algebra of all Fourier multipliers on a Banach function space $$X(\mathbb {R})$$ such that the Hardy–Littlewood maximal operator is bounded on $$X(\mathbb {R})$$ and on its associate space $$X'(\mathbb {R})$$ . For two sets $$\varPsi ,\varOmega \subset \mathcal {M}_{X(\mathbb {R})}$$ , let $$\varPsi _\varOmega$$ be the set of those $$c\in \varPsi$$ for which there exists $$d\in \varOmega$$ such that the multiplier norm of $$\chi _{\mathbb {R}\setminus [-N,N]}(c-d)$$ tends to zero as $$N\rightarrow \infty$$ . In this case, we say that the Fourier multiplier c is equivalent at infinity to the Fourier multiplier d. We show that if $$\varOmega$$ is a unital Banach subalgebra of $$\mathcal {M}_{X(\mathbb {R})}$$ consisting of nice Fourier multipliers (for instance, continuous or slowly oscillating in certain sense) and $$\varPsi$$ is an arbitrary unital Banach subalgebra of $$\mathcal {M}_{X(\mathbb {R})}$$ , then $$\varPsi _\varOmega$$ is a also a unital Banach subalgebra of $$\mathcal {M}_{X(\mathbb {R})}$$ .