Algebra of convolution type operators with continuous data on Banach function spaces

We show that if the Hardy–Littlewood maximal operator is bounded on a reflexive Banach function space $X(\mathbb R)$ and on its associate space $X’(\mathbb R)$, then the space $X(\mathbb R)$ has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in $X(\mathbb R)$, we prove that the ideal of compact operators ${\cal K}(X(\mathbb R))$ on the space $X(\mathbb R)$ is contained in the Banach algebra generated by all operators of multiplication $aI$ by functions $a\in C(\dot {\mathbb R})$, where $\dot{ \mathbb R}=\mathbb R\cup\{\infty\}$, and by all Fourier convolution operators $W^0(b)$ with symbols $b\in C_X(\dot{ \mathbb R})$, the Fourier multiplier analogue of $C(\dot{ \mathbb R})$.