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 the natural numbers. This Banach space has the form C₀(K_A) for a locally compact Hausdorff space K_A 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(K_A) is as small as possible in the precise sense that every bounded linear operator on C₀(K_A) 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(K_A) has the
fewest possible decompositions: whenever C₀(K_A) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C₀(K_A) 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(K_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 N x 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.