




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 ω _{1}unital C*algebras have <<increasing
approximate units, extending a classical result for σunital 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(l _{2}(2 ^{ω})) but,
even when the continuum hypothesis fails, we can still construct such a
C*subalgebra of B(l _{2}(ω _{1})) by using a Canadian tree.
We further show that the existence of such a C*subalgebra of B(l _{2}) 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(l_{2}) 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(l _{2}(c)) by K(l _{2}),
where c denotes the cardinality of continuum, i.e., a C*subalgebra A of B(l _{2}) satisfying the short exact sequence
0→K(l_{2})→^{i}A→K(l_{2}(c))→0,
where i[K(l _{2})] 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(l _{2}), does not have to be stable.
This construction can be considered as a noncommutative version
of Mrówka's Ψspace; a space whose one point compactification
equals to its CechStone 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(l _{2}), but the almost matrix units corresponding to the matrix units in K(l _{2}(c)) must be constructed with extra care.
 














Saeed Ghasemi, Piotr Koszmider, Noncommutative CantorBendixson derivatives and scattered C*algebras 










We analyze the sequence obtained by consecutive applications of the
CantorBendixson derivative for a noncommutative scattered C*algebra
A, using the ideal I ^{At}(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 A _{α}, along an uncountable limit ordinal λ, such that
A _{α+1}/A _{α} is *isomorphic to the algebra of all compact
operators on a separable Hilbert space and A _{α+1} is
σunital and stable for each α<λ, but A is not
stable and where all ideals of A are of the form
A _{α}. 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 A _{α}s and A satisfy the following version of the
definition of an AF algebra: any finite subset can be approximated from a
finitedimensional subalgebra. Two more complex constructions based on the
language developed in this paper are presented in separate papers.
 














Antonio Aviles, Piotr Koszmider, A 1separably injective space that does not contain l_{∞} 










We study the ω _{2}subsets of tightly σfiltered Boolean algebras and,
as an application, we show the consistency of the existence
of a Banach space that is 1separably 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.
 














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 settheoretic
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 antiRamsey 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
 














Piotr Koszmider; On constructions with 2cardinals











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 2cardinals. 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 Δproperty 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 tailormade for
crosscultural collaboration...
such efforts cannot fail to enrich both mathematical cultures"
 T. Gowers 
The two cultures of mathematics
