Algebras of Convolution Type Operators with Continuous Data do Not Always Contain All Rank One Operators

Abstract

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 \(1<p<\infty \).