On large indecomposable Banach spaces

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular)

A C(K) Banach space which does not have the Schroeder-Bernstein property

We construct a totally disconnected compact Hausdorff space N which has clopen subsets M included in L included in N such that N is homeomorphic to M and hence C(N) is isometric as a Banach space to C(M) but C(N) is not isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces of the form C(K). N is obtained as a particular compactification of the pairwise disjoint union of a sequence of Ks for which C(K)s have few operators

On constructions with 22-cardinals

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat simplified morasses called $2$-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. A new result which we obtain as a side product is the consistency of the existence of a function $f:[\lambda^{++}]^2\rightarrow[\lambda^{++}]^{\leq\lambda}$ with the appropriate $\lambda^+$-version of property $\Delta$ for regular $\lambda\geq\omega$ satisfying $\lambda^{<\lambda}=\lambda$.Comment: Minor correction

Some topological invariants and biorthogonal systems in Banach spaces

We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following associations between compact topological spaces and Banach spaces: a compact K induces a Banach space C(K) of real valued continuous functions on K with the supremum norm; a Banach space X induces a compact space - the dual ball with the weak* topology. We inquire on how topological invariants on K and the dual ball are linked to the sizes of biorthogonal systems and their versions in C(K) and X respectively. We gather folkloric facts and survey recent results like that of Lopez-Abad and Todorcevic that it is consistent that there is a Banach space X without uncountable biorthogonal systems such that the spread of the dual ball is uncountable or that of Brech and Koszmider that it is consistent that there is a compact space where spread of the square of K ic countable but C(K) has uncountable biorthogonal systems