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

We consider the closed linear subspace X(A) of the Banach space of real  bounded sequences (l_infinity) generated  by sequences converging to zero (c_0) and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N.  This Banach space X(A) has the form C_0(K_A) for a locally compact  Hausdorff  space K_A that  is  known  under  many  names,  including psi-space and Isbell–Mrówka space of A. We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on X(A) is as small as possible in the precise sense that every bounded linear operator on X(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 X(A) has  the fewest  possible decompositions:  whenever X(A) is written  as the direct sum of two infinite-dimensional Banach spaces Y and Z, either Y is isomorphic to X(A) and Z 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 X(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 NxN 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.

Based on the article: Koszmider, Piotr; Laustsen, Niels Jakob; A Banach space induced by an almost disjoint family, admitting only few operators and decompositions. Adv. Math. 381 (2021), 107613.  Available also at matharxiv: arxiv.org/pdf/2003.03832.pdf