Piotr Koszmider mail: piotr.koszmider@impan.pl

Tristan Bice, Piotr Koszmider; C*-algebras with and without <<-increasing approximate units
For elements a, b of a C*-algebra we denote a=ab by a<< b. We show that all ω1-unital C*-algebras have <<-increasing approximate units, extending a classical result for σ-unital C*-algebras. We also construct (in ZFC) the first example of a C*-algebra with no <<-increasing approximate unit. This example is a C*-subalgebra of B(l2(2ω)) but, even when the continuum hypothesis fails, we can still construct such a C*-subalgebra of B(l21)) by using a Canadian tree. We further show that the existence of such a C*-subalgebra of B(l2) is independent of ZFC.

These examples are, by necessity, not approximately finite dimensional (AF), but they are still scattered and so locally finite dimensional (LF) in the sense of Farah and Katsura. It follows that there are scattered C*-algebras which are not AF and that the existence of an LF but not AF subalgebra of B(l2) is independent of ZFC

Saeed Ghasemi, Piotr Koszmider; An extension of compact operators by compact operators with no nontrivial multipliers
We construct an essential extension of K(l2(c)) by K(l2), where c denotes the cardinality of continuum, i.e., a C*-subalgebra A of B(l2) satisfying the short exact sequence
where i[K(l2)] is an essential ideal of A such that the algebra of multipliers M(A) of A is equal to the unitization of A. In particular A is not stable which sheds light on permanence properties of the stability in the nonseparable setting. Namely, an extension of a nonseparable algebra of compact operators, even by K(l2), does not have to be stable. This construction can be considered as a noncommutative version of Mrówka's Ψ-space; a space whose one point compactification equals to its Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of N. The role of the almost disjoint family is played by an almost orthogonal family of projections in B(l2), but the almost matrix units corresponding to the matrix units in K(l2(c)) must be constructed with extra care.
Saeed Ghasemi, Piotr Koszmider, Noncommutative Cantor-Bendixson derivatives and scattered C*-algebras
We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered C*-algebra A, using the ideal IAt(A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C*-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context.
This leads to some new constructions of noncommutative scattered C*-algebras and new open problems. In particular, we construct a type I C*-algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that Aα+1/Aα is *-isomorphic to the algebra of all compact operators on a separable Hilbert space and Aα+1 is σ-unital and stable for each α<λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C*-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rordam in the nonseparable case. All the above C*-algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers.

Antonio Aviles, Piotr Koszmider, A 1-separably injective space that does not contain l
We study the ω2-subsets of tightly σ-filtered Boolean algebras and, as an application, we show the consistency of the existence of a Banach space that is 1-separably injective but does not contain any isomorphic copy of l.

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.

Accepted to Advances in Mathematics

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.

Accepted to Isreal Journal of Mathematics

"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