Calkin images of Fourier convolution operators with slowly oscillating symbols

Let $Φ$ be a $C^*$-subalgebra of $L^\infty(\mathbb{R})$ and $SO_{X(\mathbb{R})}^\diamond$ be the Banach algebra of slowly oscillating Fourier multipliers on a Banach function space $X(\mathbb{R})$. We show that the intersection of the Calkin image of the algebra generated by the operators of multiplication $aI$ by functions $a\inΦ$ and the Calkin image of the algebra generated by the Fourier convolution operators $W^0(b)$ with symbols in $SO_{X(\mathbb{R})}^\diamond$ coincides with the Calkin image of the algebra generated by the operators of multiplication by constants.

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