




T. Kania, P. Koszmider, N. J. Laustsen,
Banach spaces whose algebra of bounded operators has the integers as their K_{0}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 K _{0}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 K _{0}(B(C([0,w _{1}]))) is Z, where C([0,w _{1}]) denotes the Banach space of scalarvalued,
continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal w _{1}, 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 scalarvalued,
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,w _{1}]
Using this result, we deduce that a Banach space which is a quotient of C
can either be embedded in a Hilbertgenerated Banach space, or it is isomorphic
to the direct sum of C and a subspace of a Hilbertgenerated
Banach space. Moreover, we obtain a list of eight equivalent conditions describing the
LoyWillis 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 settheoretic 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 tailormade for
crosscultural collaboration...
such efforts cannot fail to enrich both mathematical cultures"
 T. Gowers 
The two cultures of mathematics
