论文标题
免费的Banach格子
Free Banach lattices
论文作者
论文摘要
我们研究了Banach Space $ e $上的免费$ P $ -Convex Banach Lattice $ fbl^{（p）} [e] $的结构。在回顾了为什么存在这样的免费晶格并给出方便的功能表现形式之后，我们将研究重点放在运算符$ t的属性$ t：e \ rightarrow f $之间如何转移到相关的晶格同质$ \ operline $ \ operline {t} {t}：fbl^{（p）} [e] [e] [e] [e] [e] [e] \ rightROW fbl fbl^$ fbl^=（p）。当操作员$ t $是同构嵌入时，特别考虑了这种情况，这导致我们检查运算符的扩展属性到$ \ ell_p $中，以及几个经典的Banach空间属性，例如G.T.空间。提供了免费的Banach晶格的基本序列和sublattices的详细研究。此外，我们开始在$ e $的Banach Space Properties和$ FBL^{（P）} [E] $的Banach Lattice Properties之间构建字典。特别是，我们表征了$ \ ell_1 $ in $ fbl^{（p）} [e] $的晶格副本的存在，并显示$ fbl [e] $具有上限$ p $ - 仅在$ id_ id_ iD_ {e^*} $ IS $（q，q，q，q，1）$ - sumper-shimp shimp-shimp-shimp-shimp shimp-shimp shimp shimp shimp （$ \ frac {1} {p}+\ frac {1} {q} = 1 $）。我们还强调了$ fbl^{（p）} $ - 空间之间的显着差异，具体取决于$ p $是有限的还是无限的。例如，我们表明$ fbl^{（\ infty）} [e] $是晶格等距到$ fbl^{（\ infty）} [f] $，每当$ e $ and $ e $和$ f $都具有单调有限尺寸分解时等距。
We investigate the structure of the free $p$-convex Banach lattice $FBL^{(p)}[E]$ over a Banach space $E$. After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how properties of an operator $T:E\rightarrow F$ between Banach spaces transfer to the associated lattice homomorphism $\overline{T}:FBL^{(p)}[E]\rightarrow FBL^{(p)}[F]$. Particular consideration is devoted to the case when the operator $T$ is an isomorphic embedding, which leads us to examine extension properties of operators into $\ell_p$, and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is provided. In addition, we begin to build a dictionary between Banach space properties of $E$ and Banach lattice properties of $FBL^{(p)}[E]$. In particular, we characterize the existence of lattice copies of $\ell_1$ in $FBL^{(p)}[E]$ and show that $FBL[E]$ has an upper $p$-estimate if and only if $id_{E^*}$ is $(q,1)$-summing ($\frac{1}{p}+\frac{1}{q}=1$). We also highlight the significant differences between $FBL^{(p)}$-spaces depending on whether $p$ is finite or infinite. For example, we show that $FBL^{(\infty)}[E]$ is lattice isometric to $FBL^{(\infty)}[F]$ whenever $E$ and $F$ have monotone finite dimensional decompositions, while, on the other hand, when $p<\infty$ and $E^*$ is smooth, $FBL^{(p)}[E]$ determines $E$ isometrically.
