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

Abstract

We consider the closed subspace of ℓ∞ generated by c0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C0(KA) for a locally compact Hausdorff space KA that is known under many names, including Ψ-space and Isbell–Mrówka space.We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on C0(KA) is as small as possible in the precise sense that every bounded linear operator on C0(KA) is the sum of a scalar multiple of the identity and an operator that factors through c0 (which in this case is equivalent to having separable range). This implies that C0(KA) has the fewest possible decompositions: whenever C0(KA) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C0(KA) and Y to c0, 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 C0(KA) 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 N×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.