A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

Abstract

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 $\Psi$-space and Isbell--Mr\'owka 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 Mr\'owka 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.