Koszmider, Piotr; Laustsen, Niels Jakob A Banach space induced by an almost disjoint family, admitting only few operators and decompositions. (English) Zbl 07319242 Adv. Math. 381, Article ID 107613, 40 p. (2021). Summary: 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 \(\Psi\)-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. Cited in 6 Documents MSC: 47L10 Algebras of operators on Banach spaces and other topological linear spaces 46E15 Banach spaces of continuous, differentiable or analytic functions 54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.) 46B26 Nonseparable Banach spaces 46H40 Automatic continuity 47B38 Linear operators on function spaces (general) 47L20 Operator ideals 54D45 Local compactness, \(\sigma\)-compactness Keywords:Banach space of continuous functions; \(\Psi\)-space; almost disjoint family; partitioner; perfect set property; closed operator ideal PDFBibTeX XMLCite \textit{P. Koszmider} and \textit{N. J. Laustsen}, Adv. Math. 381, Article ID 107613, 40 p. 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