论文标题
不变的阿贝尔群体捕获了巴纳克空间在其第二个偶数中的可补充性
Invariant means on Abelian groups capture complementability of Banach spaces in their second duals
论文作者
论文摘要
令$ x $为Banach空间。然后，$ x $在双交易$ x^{**} $中得到补充，并且仅当存在不变的均值$ \ ell_ \ ell_ \ infty（g，x）\ x $相对于免费的Abelian $ g $等级等于$ x^{****} $，并且仅在excant上添加了expariant nyvars，等于$ x^{**} $的基数等级等于$ x^{**} $ $ x^{**} $。考虑到bustos domecq =和第二名作者的先前结果，这有所改善，其中考虑了某些基本的基数等于$ x^{**} $的基数，并回答了J.M.F.的问题。卡斯蒂略（私人通信）。在前往主要结果证明的途中，我们赋予了无限维矢量空间的所有有限维子空间的家族，其结构具有自由交换性的单体结构，其属性具有两个子空间的乘积包含相应的子空间，这本身可能是感兴趣的。
Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.
