This is an announcement for the paper "Quasi-Banach spaces of almost universal disposition" by Felix Cabello Sanchez, Joanna Garbulinska, and Wieslaw Kubis.
Abstract: We show that for each $p\in(0,1]$ there exists a separable $p$-Banach space $\mathbb G_p$ of almost universal disposition, that is, having the following extension property: for each $\epsilon>0$ and each isometric embedding $g:X\to Y$, where $Y$ is a finite dimensional $p$-Banach space and $X$ is a subspace of $\mathbb G_p$, there is an $\epsilon$-isometry $f:Y\to \mathbb G_p$ such that $x=f(g(x))$ for all $x\in X$. Such a space is unique, up to isometries, does contain an isometric copy of each separable $p$-Banach space and has the remarkable property of being ``locally injective'' amongst $p$-Banach spaces. We also present a nonseparable generalization which is of universal disposition for separable spaces and ``separably injective''. No separably injective $p$-Banach space was previously known for $p<1$.
Archive classification: math.FA
Mathematics Subject Classification: 46A16, 46B04
Remarks: 22 pages
Submitted from: kubis@math.cas.cz
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1309.7649
or