Banach Spaces of Almost Universal Complemented Disposition

Abstract

We first unify all notions of partial injectivity appearing in the literature -(universal) separable injectivity, (universal) ℵ-injectivity -in the notion of (α, β )-injectivity ((α, β ) λ -injectivity if the parameter λ has to be specified). Then, extend the notion of space of universal disposition to space of universal (α, β )-disposition. Finally, we characterize the 1-complemented subspaces of spaces of universal (α, β )-disposition as precisely the spaces (α, β ) 1injective. 9 DEPARTAMENTO DE MATEMATICAS, UNIVERSIDAD DE EXTREMADURA, 06071 BADAJOZ, ESPAÑA

Universal disposition is not a 3-space property

“…Observe that, even under CH, there are at least three different non isomorphic spaces of universal disposition: F 1 , F 0 (see [2]) and F(C(∆)), where C(∆) is a space of continuous functions on a compact that admits a representation 0 − −−− → c 0 − −−− → C(∆) − −−− → c 0 (ℵ 1 ) − −−− → 0 (see [14]). The spaces F 1 and F 0 are very different since F 1 is 1separably injective while F 0 is not even separably injective; F 1 is a Grothendieck space while every copy of c 0 inside F 0 is complemented.…”

“…To prove that universal disposition is not a 3-space property, we use [14, Proposition 2.1] and [14,Lemma 3.2] to get that both C(ω ω ) and c 0 are complemented in the space of universal disposition F(C(ω ω )). Therefore, multiplying adequately on the left and on the right one can obtain an exact sequence…”

“…Further aspects of spaces of universal disposition have been studied in [14,12,23]. The paper [11] introduces and studies the properties of (almost) universal complemented disposition and establishes the Kadets space [21] as a Gurariy space in a different category. The results of [11] are extended in [7] to cover the categories of p-Banach spaces but using a Fraïssé like approach in the spirit of [15].…”

“…Under CH, all spaces of separable universal disposition having density character at most c are isometric (see [1,2]) and we will denote this unique specimen by F 1 . However, there are different non isomorphic spaces of universal disposition [2,14]; and also different spaces of separable universal disposition outside CH [2].…”

“…The papers [6,4] extend the methods of [1,22] to quasi-Banach and Fréchet spaces. Further aspects of spaces of universal disposition have been studied in [14,12,23]. The paper [11] introduces and studies the properties of (almost) universal complemented disposition and establishes the Kadets space [21] as a Gurariy space in a different category.…”