Piotr Koszmider mail: P.Koszmider@impan.pl

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.

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.

T. Kania, P. Koszmider, N. J. Laustsen, Banach spaces whose algebra of bounded operators has the integers as their K0-group
Let X and Y be Banach spaces such that the ideal of operators which factor through Y has codimension one in the Banach algebra B(X) of all bounded operators on X, and suppose that Y contains a complemented subspace which is isomorphic to Y+Y and that X is isomorphic to X+Z for every complemented subspace Z of Y. Then the K0-group of B(X) is isomorphic to the additive group Z of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that K0(B(C([0,w1]))) is Z, where C([0,w1]) denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal w1, endowed with the order topology.

Piotr Koszmider; Universal objects and associations between classes of Banach spaces and classes of compact spaces
In the context of classical associations between classes of Banach spaces and classes of compact Hausdorff spaces we survey known results and open questions concerning the existence and nonexistence of universal Banach spaces and of universal compact spaces in various classes. This gives quite a complex network of interrelations which quite often depend on additional set-theoretic assumptions.

Accepted to a special issue of Publications de l'Institut Mathématique (Beograd)
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"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