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, including Ψ-space and Isbell–Mrówka space.

We construct an uncountable, almost disjoint family $\mathcal{A}$ such that the algebra of all bounded linear operators on $C_0(K_{\mathcal{A}})$ is as small as possible in the precise 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 $C_0(K_{\mathcal{A}})$ is written as the direct sum of two infinite-dimensional Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, either $\mathcal{X}$ is isomorphic to $C_0(K_{\mathcal{A}})$ and $\mathcal{Y}$ 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 by Mrówka, we need to deal with $\mathbb{N}\times \mathbb{N}$ matrices rather than with the usual partitioners of an almost disjoint family. 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 van Engelen, Kunen and Miller concerning Borel subsets of the square.

我们考虑的封闭子空间 ${\ell }_{\infty }$ 由...产生 $C_0$ 以及不可数的，几乎不相交的家庭元素的特征功能 $\mathcal{一个}$ 的无限子集 $\mathbb{ñ}$。该Banach空间具有以下形式$C_0（{ķ}_{\mathcal{一个}}）$ 用于局部紧凑的Hausdorff空间 ${ķ}_{\mathcal{一个}}$ 这就是众所周知的许多名称，包括Ψ-空间和Isbell-Mrówka空间。

我们建立了一个无法计数，几乎不相交的家庭 $\mathcal{一个}$ 使得所有有界线性算子的代数 $C_0（{ķ}_{\mathcal{一个}}）$ 从精确的意义上讲，它应尽可能小， $C_0（{ķ}_{\mathcal{一个}}）$ 是恒等式和运算符的标量倍数之和 $C_0$（在这种情况下，它等同于具有可分离的范围）。这意味着$C_0（{ķ}_{\mathcal{一个}}）$ 分解最少的东西：每当 $C_0（{ķ}_{\mathcal{一个}}）$ 写为两个无限维Banach空间的直接和 $\mathcal{X}$ 和 $\mathcal{ÿ}$，或者 $\mathcal{X}$ 同构 $C_0（{ķ}_{\mathcal{一个}}）$ 和 $\mathcal{ÿ}$ 至 $C_0$， 或相反亦然。这些结果改进了第一个具名作者的先前工作，在该工作中需要额外的集合理论假设。我们还将讨论这些结果对Banach空间上所有有界线性算子的代数的影响$C_0（{ķ}_{\mathcal{一个}}）$ 关于封闭理想的格，特征和同构的自动连续性。

要像Mrówka在一个几乎不相交的家庭的经典构造中那样，为Borel集使用完美的集属性，我们需要处理 $\mathbb{ñ}\times \mathbb{ñ}$矩阵，而不是几乎不相交的家庭的普通分隔线。这种非可交换的设置要求新的思想受到紧致和弱紧致算子理论的启发，并由于范·恩格伦，库恩和米勒关于正方形的Borel子集而采用提取原理。