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

Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X^′(R). The algebra C_X(R˙) of continuous Fourier multipliers on X(R) is defined as the closure of the set of continuous functions of bounded variation on R˙ = R∪ { ∞} with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X(R) is reflexive, then the ideal of compact operators is contained in the Banach algebra A_X(R) generated by all multiplication operators aI by continuous functions a∈ C(R˙) and by all Fourier convolution operators W(b) with symbols b∈ C_X(R˙). We show that there are separable and non-reflexive Banach function spaces X(R) such that the algebra A_X(R) does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces L^p,1(R) with 1 < p< ∞.