Discovering the algebraic structure on the metric injective envelope of a real Banach space (Q1361383)

By \textit{H. B. Cohen} [Bull. Am. Math. Soc. 70, 723-726 (1964; Zbl 0124.06505)], every real Banach space \(X\) admits an injective envelope, i.e. there exists a real injective Banach space \(\varepsilon\ell X\) and a linear isometry \(e: X\to \varepsilon\ell X\) such that the smallest injective subspace of \(\varepsilon\ell X\) containing \(e(X)\) is \(\varepsilon\ell X\) itself. In the present paper, the authors give an alternative construction of an injective envelope by introducing the structure of a Banach space on the injective metric envelope \(\varepsilon mX\) of \(X\), defined by Isbell. In doing this, they answer a question of \textit{J. R. Isbell} [J. Math. Anal. Appl. 27, 516-518 (1969; Zbl 0206.42201)].