Let \(X(\mathbb {R})\) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on \(X(\mathbb {R})\) and on its associate space \(X'(\mathbb {R})\). The algebra \(C_X(\mathbf{\dot{\mathbb {R}}})\) of continuous Fourier multipliers on \(X(\mathbb {R})\) is defined as the closure of the set of continuous functions of bounded variation on \(\mathbf{\dot{\mathbb {R}}}=\mathbb {R}\cup \{\infty \}\) with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] 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 \(a\in C(\mathbf{\dot{\mathbb {R}}})\) and by all Fourier convolution operators \(W^0(b)\) with symbols \(b\in C_X(\mathbf{\dot{\mathbb {R}}})\). 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 \(1<p<\infty \).

令\（X（\ mathbb {R}）\）为可分离的Banach函数空间，这样Hardy-Littlewood最大运算符就以\（X（\ mathbb {R}）\）及其关联空间\（X '（\ mathbb {R}）\）。\（X（\ mathbb {R}）\）上连续傅立叶乘法器的代数\ {C_X（\ mathbf {\ dot {\ mathbb {R}} }）\）被定义为连续函数集的闭包乘数范数对\（\ mathbf {\ dot {\ mathbb {R}}} = \ mathbb {R} \ cup \ {\ infty \} \的有界变化的影响。Yu。C. Fernandes证明了这一点。卡洛维奇和第一作者[11]认为，如果空格\（X（\ mathbb {R}）\）如果是自反的，则紧致算子的理想包含在由所有乘法算子aI由连续函数\（a \ in C生成）的Banach代数\（\ mathcal {A} _ {X（\ mathbb {R}）} \\）中（\ mathbf {\点{\ mathbb {R}}}）\）和由所有傅立叶卷积算\（W ^ 0（b）中\）用符号\（b \在C_X（\ mathbf {\点{\ mathbb {R}}}）\）。我们证明了存在可分离且非自反的Banach函数空间\（X（\ mathbb {R}）\）使得代数\（\ mathcal {A} _ {X（\ mathbb {R}）} \）可以不包含所有排名第一的运营商。特别地，这种情况发生在Lorentz空间的情况下\（L ^ {P，1}（\ mathbb {R}）\）与\（1 <p <\ infty \）。