Calkin images of Fourier convolution operators with slowly oscillating symbols

Abstract

Let $\Phi$ be a $C^{\ast }$-subalgebra of $L^{\infty }(\mathbb{R})$ and $S{O}_{X(\mathbb{R})}^{\diamond }$ 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 $a\in \Phi$ and the Calkin image of the algebra generated by the Fourier convolution operators $W^0(b)$ with symbols in $S{O}_{X(\mathbb{R})}^{\diamond }$ coincides with the Calkin image of the algebra generated by the operators of multiplication by constants.

To appear in the Proceedings of IWOTA 2019. arXiv admin note: text overlap with arXiv:2007.10266