论文标题
一个几乎不相交的家族引起的Banach空间，仅承认少数操作员和分解
A Banach space induced by an almost disjoint family, admitting only few operators and decompositions
论文作者
论文摘要
我们考虑$ c_0 $生成的$ \ ell_ \ infty $的封闭子空间以及一个不可容纳的，几乎不连接的家族$ \ mathcal的元素的特征功能，$ \ mathbb n $的无限子集。此Banach空间的形式具有$ C_0（k _ {\ Mathcal a}）$，用于本地紧凑的Hausdorff Space $ k _ {\ Mathcal a} $，以许多名称（例如$ψ$ space）和isBell-mrówkaSpace所知。 我们构建了一个不可容纳的，几乎不连贯的家族$ {\ Mathcal a} $，以使所有有界线性运算符的Banach代数在$ C_0（K _ {\ Mathcal a}）上$很小，可以在$ C_0（k _ {k _ {\ Mathcal a} actection in a a} a} a} a} a a} a a i is a a a i s a a a} a a i is a a a i a a} a的限制线性操作员时，a i is a a} a} a} a} a} a} $ C_0 $（在这种情况下，等效于具有可分离范围）。这意味着$ c_0（k _ {\ mathcal a}）$具有最少的分解：每当$ c_0（k _ {\ mathcal a}）= x \ oplus y $带有$ dim（{x}） $ c_0（k _ {\ mathcal a}）$和$ {y} $ to $ c_0 $，反之亦然。这些结果改善了第一位命名作者的先前工作，其中需要额外的设定理论假设。我们还讨论了这些结果对Banach空间上所有有界线性运算符的代数的后果$ C_0（K _ {\ Mathcal a}）$关于封闭的理想，字符和同源性自动连续性的晶格。 为了利用Borel套装的完美套装属性，就像在Mrówka几乎不相关家族的经典构造中，我们需要处理$ \ Mathbb N \ Times \ Times \ Mathbb n $ Matrices，而不是与通常的分区者。这种非公共设置需要由紧凑和紧凑的操作员理论启发的新想法，以及由于F. van Engelen，K。K. Kunen和A. Miller而引起的提取原则，就广场的Borel子集而言。
We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form $C_0(K_{\mathcal A})$ for a locally compact Hausdorff space $K_{\mathcal A}$ that is known under many names, such as $Ψ$-space and Isbell--Mrówka space. We construct an uncountable, almost disjoint family ${\mathcal A}$ such that the Banach algebra of all bounded linear operators on $C_0(K_{\mathcal A})$ is as small as possible in the sense that every bounded linear operator on $C_0(K_{\mathcal A})$ is the sum of a scalar multiple of the identity and an operator that factors through $c_0$ (which in this case is equivalent to having separable range). This implies that $C_0(K_{\mathcal A})$ has the fewest possible decompositions: whenever $C_0(K_{\mathcal A})=X\oplus Y$ with $dim({X})=\infty$, $dim({Y})=\infty$, either ${X}$ is isomorphic to $C_0(K_{\mathcal A})$ and ${Y}$ to $c_0$, or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space $C_0(K_{\mathcal A})$ concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms. To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family of Mrówka we need to deal with $\mathbb N \times \mathbb N$-matrices rather than with the usual partitioners. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to F. van Engelen, K. Kunen and A. Miller concerning Borel subsets of the square.
