∞(γ) in Quotients of Banach Lattices

Let E be a Banach lattice and let M be a norm-closed and Dedekind σ-complete ideal of E. If E contains a lattice-isometric copy of ∞ , then E/M contains such a copy as well, or M contains a lattice copy of ∞. This is one of the consequences of more general results presented in this paper. 1. Introduction. Let E be a locally solid linear lattice (Riesz space), for example , a Banach lattice, let M be a closed ideal of E, and let Γ be an infinite set. In [7, Theorem 1] it is proved that if E contains a lattice copy U of ∞ and M is Dedekind σ-complete, then E/M or M contains such a copy as well. (Here and in what follows the term lattice copy means both lattice and topological copy, and lattice-isometric copy means both lattice and iso-metric copy.) This is a lattice-topological version of the classical by now theorem of Drewnowski and Roberts which asserts that the noncontainment of ∞ is a three-space property in the class of Banach spaces (see [3, Theorem