论文标题
三个占人数问题
Three-representation problem
论文作者
论文摘要
我们提供了先前宣布的结果的证明，该结果解决了A.〜A.〜Kirillov提出的以下问题。令$ t $为Banach Space $ G $中的有限线性运算符和$ e \ e \ subset g $的组$ \ Mathcal {G} $的介绍。然后，$ t $以自然方式生成$ e $中的$ t_1 $和$ f：f：= g/e $的$ t_2 $。除了$ t_1，t_2 $以恢复演示文稿$ t $之外，还需要哪些其他信息？在有限维（甚至在无限维度希尔伯特）的情况下，解决方案是众所周知的：一个人需要在h^1中提供群体的共同体级$ h \（\ MATHCAL {G}，HOM（f，e））$。如果子空间$ e $在$ g $中补充，则在Banach案中也是如此。但是，每个对希尔伯特一号不是同构的BANACH空间都有非补充的子空间，即使在琐碎的群体行动的情况下，也会使问题严重加剧，并且使其变得不乏味，在这种情况下，它归结为所谓的三个空间问题。这解释了我们选择的标题。作者在1976年宣布了上述问题的解决方案，但由于非数学原因，尚未提供完整的证据。本文包含Banach Space的类别\ TextBf {Ban}的函数$ ext^1 $的证据以及一些相关考虑因素。
We provide the proof of a previously announced result that resolves the following problem posed by A.~A.~Kirillov. Let $T$ be a presentation of a group $\mathcal{G}$ by bounded linear operators in a Banach space $G$ and $E\subset G$ be a closed invariant subspace. Then $T$ generates in the natural way presentations $T_1$ in $E$ and $T_2$ in $F:=G/E$. What additional information is required besides $T_1, T_2$ to recover the presentation $T$? In finite-dimensional (and even in infinite dimensional Hilbert) case the solution is well known: one needs to supply a group cohomology class $h\in H^1(\mathcal{G},Hom(F,E))$. The same holds in the Banach case, if the subspace $E$ is complemented in $G$. However, every Banach space that is not isomorphic to a Hilbert one has non-complemented subspaces, which aggravates the problem significantly and makes it non-trivial even in the case of a trivial group action, where it boils down to what is known as the three-space problem. This explains the title we have chosen. A solution of the problem stated above has been announced by the author in 1976, but the complete proof, for non-mathematical reasons, has not been made available. This article contains the proof, as well as some related considerations of the functor $Ext^1$ in the category \textbf{Ban} of Banach spaces.
