## Infinite Asymptotic Games and $(*)$-Embeddings of Banach Spaces

### Volume 60 / 2012

#### Abstract

We use methods of infinite asymptotic games to characterize subspaces of Banach spaces with a finite-dimensional decomposition (FDD) and prove new theorems on operators. We consider a separable Banach space $X$, a set $\mathcal{S}$ of sequences of finite subsets of $X$ and the $\mathcal{S}$-game. We prove that if $\mathcal{S}$ satisfies some specific stability conditions, then Player I has a winning strategy in the $\mathcal{S}$-game if and only if $X$ has a skipped-blocking decomposition each of whose skipped-blockings belongs to $\mathcal{S}$. This result implies that if $T$ is a $(*)$-embedding of $X$ (a 1-1 operator which maps the balls of subspaces with an FDD to weakly $G_{\delta}$ sets), then, for every $n\geq 4$, there exist $n$ subspaces of $X$ with an FDD that generate $X$ and the restriction of $T$ to each of them is a semi-embedding under an equivalent norm. We also prove that $X$ does not contain isomorphic copies of dual spaces if and only if every $(*)$-embedding defined on $X$ is an isomorphic embedding. We also deal with the case where $X$ is non-separable, reaching similar results.