  Let $X(\mathbb{R})$ be a separable Banach function space such that the
Hardy-Littlewood maximal operator is bounded $X(\mathbb{R})$ and on its
associate space $X'(\mathbb{R})$. The algebra $C_X(\dot{\mathbb{R}})$ of
continuous Fourier multipliers on $X(\mathbb{R})$ is defined as the closure of
the set of continuous functions of bounded variation on
$\dot{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ with respect to the multiplier
norm. It was proved by C. Fernandes, Yu. Karlovich and the first author
\cite{FKK19} that if the space $X(\mathbb{R})$ is reflexive, then the ideal of
compact operators is contained in the Banach algebra
$\mathcal{A}_{X(\mathbb{R})}$ generated by all multiplication operators $aI$ by
continuous functions $a\in C(\dot{\mathbb{R}})$ and by all Fourier convolution
operators $W^0(b)$ with symbols $b\in C_X(\dot{\mathbb{R}})$. We show that
there are separable and non-reflexive Banach function spaces $X(\mathbb{R})$
such that the algebra $\mathcal{A}_{X(\mathbb{R})}$ does not contain all rank
one operators. In particular, this happens in the case of the Lorentz spaces
$L^{p,1}(\mathbb{R})$ with $1<p<\infty$.
