Adding an identity to a Banach lattice algebra: a look at a Wickstead’s counter-example from a norm-free point of view

Abstract

Recently, Wickstead investigated the long-standing problem of adding an identity to a non-unital Banach lattice algebra. In this regard, he proved that, in the category whose objects are unital Banach lattice algebras and morphisms are identity preserving algebra and lattice homomorphisms, there is no reflection for $c_{0}$ in which $c_{0}$ can be embedded as an algebra and order ideal. The main purpose of this note is to describe an alternative category in which a suitable reflection of $c_{0}$ can be located, producing a satisfactory lattice unitization of $c_{0}$ with $c_{0}$ as an algebra and order ideal.