On almost isometric ideals in Banach spaces

In this note we study the notion of almost isometric ideals recently introduced in Abrahamsen et al. (Glasg Math J 56:395–407, 2014) for separable Banach spaces. We show that a closed subspace of the Gurariy space that is an almost isometric ideal is itself the Gurariy space. In this space we show that all infinite dimensional M-ideals are almost isometric ideals. In c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0$$\end{document} we show that any finite codimensional proximinal ideal is an almost isometric ideal. We solve in the negative the 3-space problem for almost isometric ideals. We also give an example to show that being an almost isomeric ideal is not preserved by spaces of vector-valued continuous functions on a compact set. We show that any separable L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-predual space with a non-separable dual, has an ideal that is not an almost isometric ideal. We also study properties of almost isometric ideals that are hyperplanes.