Piotr Koszmider mail: P.Koszmider@impan.pl

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.

T. Kania, P. Koszmider, N. J. Laustsen, A weak*-topological dichotomy with applications in operator theory
Denote by K the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C be the Banach space of scalar-valued, continuous functions which are defined on K and vanish eventually. We show that a weakly* compact subset of the dual space of C is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval [0,w1] Using this result, we deduce that a Banach space which is a quotient of C can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C and a subspace of a Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent conditions describing the Loy--Willis ideal, which is the unique maximal ideal of the Banach algebra of bounded, linear operators on C. As a consequence, we find that this ideal has a bounded left approximate identity, thus solving a problem left open by Loy and Willis, and we give new proofs, in some cases of stronger versions, of several known results about the Banach space C and the operators acting on it.

Accepted to Transactions of the London Mathemtical Society (a new journal, starting online on 22.04.2014)
For preprint click on the icon of the journal


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)
For preprint click on the icon of the journal


"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