Piotr Koszmider mail: P.Koszmider@impan.pl

Piotr Koszmider, Saharon Shelah, Michał Świętek, There is no bound on sizes of indecomposable Banach spaces
Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces, i.e., such that whenever they are decomposed as X⊕Y, then one of the closed subspaces X or Y must be finite dimensional. It requires alternative techniques compared to those which were initiated by Gowers and Maurey or Argyros with the coauthors. This is because hereditarily indecomposable Banach spaces always embed into l and so their density and cardinality is bounded by the continuum and because dual Banach spaces of densities bigger than continuum are decomposable by a result due to Heinrich and Mankiewicz. The obtained Banach spaces are of the form C(K) for some compact connected Hausdorff space and have few operators in the sense that every linear bounded operator T on C(K) for every f in C(K) satisfies T(f)=gf+S(f) where g is in C(K) and S is weakly compact or equivalently strictly singular. In particular, the spaces carry the structure of a Banach algebra and in the complex case even the structure of a C*-algebra.

Tristan Bice, Piotr Koszmider, A note on the Akemann-Doner and Farah-Wofsey constructions
We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable C*-algebra A with only separable commutative C*-subalgebras. We also extend a result of Farah and Wofsey's, constructing ℵ1 commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin's almost disjoint family construction.

Accepted to Proceedings of the AMS

Piotr Koszmider, Uncountable equilateral sets in Banach spaces of the form C(K)
The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin's axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε>0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis, G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.

Piotr Koszmider, On the problem of compact totally disconnected reflection of nonmetrizability
We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L of K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

Accepted to a special issue of Topology and Applications

Piotr Koszmider, Cristóbal Rodriguez-Porras, On automorphisms of the Banach space l/c0
We investigate Banach space automorphisms of l/c0 focusing on the possibility of representing their fragments of the form
TB,A:l(A)/c0(A) → l(B)/c0(B)

for A,B infinite subsets of N by means of linear operators from l(A) into l(B), infinite A×B-matrices, continuous maps from B* into A*, or bijections from B to A. This leads to the analysis of general linear operators on l/c0.
We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for other give examples showing that it is impossible. In particular, we show that there are automorphisms of l/c0 which cannot be lifted to operators on l and assuming OCA+MA we show that every automorphism of l/c0 with no fountains or with no funnels is locally, i.e., for some infinite A,B?N as above, induced by a bijection from B to A. This additional set-theoretic assumption is necessary as we show that the continuum hypothesis implies the existence of counterexamples of diverse flavours. However, many basic problems, some of which are listed in the last section, remain open.

Accepted to Fundamenta Mathematicae

Leandro Candido, Piotr Koszmider; On complemented copies of c01) in C(Kn) spaces
Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kn) or equivalently the n-fold injective tensor product of the C(K)s or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c01) in C(Kn) under the hypothesis that C(K) contains an isomorphic copy of c01). This is related to the results of E. Saab and P. Saab that the injective tensor product of X and Y contains a complemented copy of c0, if one of the infinite dimensional Banach spaces X or Y contains a copy of c0 and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if C(K) has density ω1 and contains a copy of c01), then C(K×K) contains a complemented copy c01).
The main result is that under the assumption of the club principle for every n in N there is a compact Hausdorff space Kn of weight ω1 such that C(Kn) is Lindelof in the weak topology, C(Kn) contains a copy of c01), C(Knn) does not contain a complemented copy of c01) while C(Knn+1) does contain a complemented copy of c01). This shows that additional set-theoretic assumptions in Galego and Hagler's nonseparable version of Cembrano and Freniche's theorem are necessary as well as clarifies in the negative direction the matter unsettled in a paper of Dow, Junnila and Pelant whether half-pcc Banach spaces must be weakly pcc.

Piotr Koszmider; On constructions with 2-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 with the appropriate version Δ-property for arbitrarily big cardinals.

Accepted to a special issue of Archive for Mathematical Logic

"Unfortunately, it is also difficult to reach a level of understanding where one can appreciate the essentailly combinatorial nature of the underlying problem. Such a situation is tailor-made for cross-cultural collaboration... such efforts cannot fail to enrich both mathematical cultures"

    -- T. Gowers --    The two cultures of mathematics