Piotr Koszmider mail: piotr.koszmider@impan.pl

"An artist is a person who knows how to make puzzles out of solutions."

    --Karl Kraus --   

Piotr Koszmider; On constructions with 2-cardinals; Arch. Math. Logic 56 (2017), no. 7-8, 849–876.
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.

Tristan Bice, Piotr Koszmider, A note on the Akemann-Doner and Farah-Wofsey constructions, Proc. Amer. Math. Soc. 145 (2017), no. 2, 681–687.
We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable C*-algebra A with only separable commutative C*-subalgebras. We also extend a result of Farah and Wofsey's, constructing ℵ1 commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin's almost disjoint family construction.

Piotr Koszmider, Cristóbal Rodriguez-Porras, On automorphisms of the Banach space l/c0, Fund. Math. 235 (2016), no. 1, 49–99.
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.

Koszmider, Piotr; On the problem of compact totally disconnected reflection of nonmetrizability. Topology Appl. 213 (2016), 154–166
We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L of K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

Leandro Candido, Piotr Koszmider; On complemented copies of c01) in C(Kn) spaces, Studia Math. 233 (2016), no. 3, 209–226.
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.

Christina Brech, Piotr Koszmider, An isometrically universal Banach space induced by a non-universal Boolean algebra, Proc. Amer. Math. Soc. 144 (2016), no. 5, 2029–2036
Given a Boolean algebra A, we construct another Boolean algebra B with no uncountable well-ordered chains such that the Banach space of real valued continuous functions C(KA) embeds isometrically into C(KB), where KA and KB are the Stone spaces of A and B respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density k, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality k. Such a phenomenon cannot happen on the level of separable Banach spaces and countable Boolean algebras. This is related to the open question if the existence of an isometrically universal Banach space and of a universal Boolean algebra are equivalent on the nonseparable level (both are true on the separable level).

Piotr Koszmider; Universal objects and associations between classes of Banach spaces and classes of compact spaces, Sbornik Radova 17(25), 93-115; Selected Topics in Combinatorial Analysis, eds. M. Kurilic and S. Todorcevic; Matematicki Institut SANU, Beograd, 2015.
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.

Webpage of the volume

T. Kania, P. Koszmider, N. J. Laustsen, Banach spaces whose algebra of bounded operators has the integers as their K0-group, 428 (2015), no. 1, 282-294
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; Trans. London Math. Soc. (2014) 1 (1): 1-28
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.

Christina Brech, Piotr Koszmider; l-sums and the Banach space l/c0; Fund. Math. 224 (2014), 175-185
We prove that the use of the Continuum Hypothesis in some results of Drewnowski and Roberts concerning the Banach space l/c0 cannot be avoided. In particular, we prove that in the Cohen model, l(c0(c)) does not embed isomorphically into l/c0 where c is the cardinality of the continuum. It follows that consistently l/c0 is not isomorphically of the form l(X) for any Banach space X.

H. G. Dales, T. Kania, T. Kochanek, P. Koszmider, N. J. Laustsen, Maximal left ideals   of the Banach algebra of bounded operators on a Banach space; Studia Math. 218 (2013), 245-286
We address the following two questions regarding the maximal left ideals of the Banach algebra B(E) of bounded operators acting on an infinite-dimensional Banach space E:
  1. Does B(E) always contain a maximal left ideal which is not finitely generated?
  2. Is every finitely-generated, maximal left ideal of B(E) necessarily of the form {T in B(E) : Tx = 0} (*) for some non-zero x E?
Since the two-sided ideal F(E) of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the second question would imply a positive answer to the first. Our main results are:
  • Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces;
  • Question (II) has a positive answer if and only if no finitely-generated, maximal left ideal of B(E) contains F(E);
  • the answer to Question (II) is positive for many, but not all, Banach spaces.

Antonio Avilés, Piotr Koszmider; A continuous image of a Radon-Nikodym compact space which is not Radon-Nikodym; Duke Math. J. 162, 12 (2013), 2285-2299.
We construct a continuous image of a Radon-Nikodym compact space which is not Radon-Nikodym compact, solving the problem posed in the 80ties by Isaac Namioka.

Antonio Avilés, Piotr Koszmider; A Banach space in which every injective operator is surjective; Bull. London Math. Soc. (2013) 45 (5): 1065-1074
We construct an infinite dimensional Banach space of continuous functions C(K) such that every one-to-one operator on C(K) is onto.

Piotr Koszmider, Saharon Shelah; Independent families in Boolean algebras with some separation properties; Algebra Universalis 69 (2013), no. 4, 305 - 312

We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has l-infinity as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.

Piotr Koszmider; On large indecomposable Banach spaces; J. Funct. Anal. 264 (2013), no. 8, 1779–1805

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular).

Jesus Ferrer, Piotr Koszmider, Wieslaw Kubis; Almost disjoint families of countable sets and separable complementation properties; J. Math. Anal. Appl. 401 (2013), no. 2, 939–949

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta KA induced by almost disjoint families A of countable subsets of uncountable sets. For these spaces, we prove among others that C(KA) has the controlled variant of the separable complementation property if and only if C(KA) is Lindelof in the weak topology if and only if KA is monolithic. We give an example of A for which C(KA) has the SCP, while KA is not monolithic and an example of a space C(KA) with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality omega-one which induce monolithic spaces of the form KA: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations.

Christina Brech, Piotr Koszmider; On universal spaces for the class of Banach spaces whose dual balls are uniform Eberlein compacts; Proc. Amer. Math. Soc. 141 (2013), 1267-1280

For k being the first uncountable cardinal w1 or k being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density k in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gateaux differentiable renorming or, equivalently, whose dual unit ball with the weak* topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, V. Zizler that assuming the continuum hypothesis, there is a universal space for all Banach spaces of density k=c=w1 which have a uniformly Gateaux differentiable renorming. Our result implies, in particular, that betaN-N may not map continuously onto a compact subset of a Hilbert space with the weak topology of density k=w1 or k=c and that a C(K) space for some uniform Eberlein compact space K may not embed isomorphically into l_infty/c_0.

Piotr Koszmider; A C(K) Banach space which does not have the Schroeder-Bernstein property; Studia Math. 212 (2012), 95-117
We construct a totally disconnected compact Hausdorff space N which has clopen subsets M included in L included in N such that N is homeomorphic to M and hence C(N) is isometric as a Banach space to C(M) but C(N) is not isomorphic to C(L). This gives two nonisomorphic Banach spaces of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces of the form C(K). N is obtained as a particular compactification of the pairwise disjoint union of a sequence of Ks for which C(K)s have few operators.

  Christina Brech, Piotr Koszmider; On universal Banach spaces of density continuum,   Israel J. Math. 190 (2012), 93–110.
We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into X (called a universal Banach space of density continuum). It is well known that l-infinity by c-zero is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density continuum. Thus, the problem of the existence of a universal Banach space of density continuum is undecidable using the usual axioms of set-theory. We also prove that it is consistent that there are universal Banach spaces of density continuum, but l-infinity by c-zero is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0,c]) into l-infinity by c-zero.

  Piotr Koszmider; Some topological invariants and biorthogonal systems in Banach   spaces, Extracta Math. 26(2) (2011), 271-294
We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following associations between compact topological spaces and Banach spaces: a compact K induces a Banach space C(K) of real valued continuous functions on K with the supremum norm; a Banach space X induces a compact space BX*, the dual ball with the weak* topology. We inquire on how topological invariants on K and BX* are linked to the sizes of biorthogonal systems and their versions in C(K) and X respectively. We gather folkloric facts and survey recent results like that of Lopez-Abad and Todorcevic that it is consistent that there is a Banach space X without uncountable biorthogonal systems such that the spread of BX* is uncountable or that of Brech and Koszmider that it is consistent that there is a compact space where spread of K2 ic countable but C(K) has uncountable biorthogonal systems.

  Christina Brech, Piotr Koszmider; On biorthogonal systems whose functionals are   finitely supported; Fund. Math. 213 (2011), 43-66

We show that for each natural n>1 it is consistent that there is a compact Hausdorff space K2n such that in C(K2n) there is no uncountable (semi)biorthogonal sequence (fi, mi) for i in w1where mi's are atomic measures with supports consisting of at most 2n-1 points of K2n, but there are biorthogonal systems (fi, mi) for i in w1where mi's are atomic measures with supports consisting of 2n points.
This complements a result of Todorcevic that Martin's axiom with the negation of CH implies that each nonseparable Banach space C(K) has an uncountable biorthogonal system where the functionals are differences of two pointwise measures. It also follows that it is consistent that the irredundance of the Boolean algebra Clop(K) or the Banach algebra C(K) for K totally disconnected can be strictly smaller than the sizes of biorthogonal systems in C(K). The compact spaces exhibit an interesting behaviour with respect to known cardinal functions: the hereditary density of the powers K{2n}k is countable up to k=n and it is uncountable (even the spread is uncountable) for k>n.

  Piotr Koszmider, Miguel Martín, Javier Merí; Isometries on extremely non-complex   Banach spaces; Journal of the Institute of Mathematics of Jussieu, 10 (2011) No. 02,   pp.325-348

We construct an example of a real Banach space whose group of surjective isometries reduces to plus or minus identity, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup. To do so, we present examples of extremely non-complex Banach spaces (i.e. spaces X such that ||Id+ T2||=1+||T2|| for every bounded linear operator T on X) which are not of the form C(K), and we study the surjective isometries on this class of Banach spaces.

  Christina Brech, Piotr Koszmider; Thin-very tall compact scattered spaces which are   hereditarily separable; Transactions of the American Mathematical Society; 363   (2011), no. 1, pp. 501 - 519

We strengthen the property Delta of a function f from pairs of omega-2 to countable subsets of omega-2 considered by Baumgartner and Shelah. This allows us to consider new type of amalgamations in the forcing used by Rabus, Juhasz and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces K as above where n-th power of K is hereditarily separable for each natural N. This serves as a counterexample concerning cardinal functions on compact spaces as well as has some applications in Banach spaces: the Banach space C(K) is an Asplund space of density aleph-2 which has no Frechet smooth renorming nor an uncountable biorthogonal system.

  Piotr Koszmider, Przemys³aw Zieliñski; Complementation and Decompositions in   some weakly Lindelof Banach spaces; J. Math. Anal. Appl. 376 (2011) 329–341
We consider the questions if a Banach space of the form C(K) of a given class (1) has a complemented copy of c0(G) for G uncountable or (2) for every c0(G) in X has a complemented c0(E) for an uncountable E in G or (3) has a decomposition X=A+B where both A and B are nonseparable. The results concern a superclass of the class of nonmerizable Eberlein compacts, namely Ks such that C(K) is Lindelof in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC plus-minus CH. If we assume the P-ideal dichotomy, for every c0(G) in C(K) there is a complemented $c0(E) for an uncountable E in G, which yields the positive answer to the remaining questions. If we assume the club axiom, then we construct a nonseparable weakly Lindelof C(K) for K of height w+1 where every operator is of the form cI+S for c real and S an operator with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelof properties of these two topologies are equivalent, many results concern also the space Cp(K).

  Piotr Koszmider; A survey on Banach spaces C(K) with few operators; RACSAM 104   (2), 2010, pp. 309 -326

We say that a Banach space C(K) has few operators if for every operator T on C(K) we have T=gI+S or T*=g*I+S where g is continuous on K, g* is Borel on K and S are weakly compact on C(K) or C*(K) respectively. Banach spaces of continuous functions with few operators provided solutions to several long standing open problems in the theory of Banach spaces. The class of spaces is being gradually illuminated and applied further in the recent work of P. Borodulin-Nadzieja, R. Fajardo, V. Ferenczi, E. Medina Galego, M. Martín, J. Merí, G. Plebanek, I. Schlackow and the author. We describe basic properties, applications and relevant open problems.

  Piotr Koszmider; On a problem of Rolewicz about Banach spaces that admit support   sets; Journal of Functional Analysis 257 (2009) pp. 2723-2741

We construct an example of a nonseparable Banach space which does not admit a support set. It is a consistent (and necessarily independent from the axioms of ZFC) example of a space $C(K)$ of continuous functions on a compact Hausdorff $K$ with the supremum norm. The construction depends on a construction of a Boolean algebra with some combinatorial properties. The space is also hereditarily Lindelof in the weak topology but it doesn't have any nonseparable subspace nor any nonseparable quotient which is a C(K) space for K dispersed.

  Istvan Juhasz, Piotr Koszmider, Lajos Soukup; A first countable, initially w1-compact,   but non-compact space; Topology and its Applications 156 (2009) pp. 1863-1879

We force a first countable, normal, locally compact, initially w1-compact but non-compact space X of size w2. The one-point compactification of X is a non-first countable compactum without any (non-trivial) converging w1-sequence.

  Artur Bartoszewicz and Piotr Koszmider; When an atomic and complete algebra of   sets is a field of sets with nowhere dense boundary; Journal of Applied Analysis 15   (2009) pp. 119-127

We consider pairs where A is an algebra of sets from some class called the class of algebras of type and where H(A) is the ideal of hereditary sets of A. We characterize which of the above pairs are topological, that is, which are fields of sets with nowhere dense boundary for some topology together with the ideal of nowhere dense sets for this topology. Making use of the Balcar-Franek theorem we construct an example of a pair with complete quotient algebra and the hull property but not topological. This countrexample, given in ZFC, provides the complete solution of a problem posed in [M. Balcerzak, A.Bartoszewicz, K.Ciesielski, Algebras with inner MB-representation, 29(1) 2003-2004, Real. Anal. Exchange.] Such an algebra was constructed in [A. Bartoszewicz, On some algebra of sets in Steprans strong-Q-sequence model, Topology Appl. 149, (2005), no. 1-3, 9--15.] under some aditional set theoretic assumption.

  Piotr Koszmider, Miguel Martín, Javier Merí; Extremely non-complex C(K) spaces;   Journal of Mathematical Analysis and Applications Volume 350, Issue 2, 15, 2009,   pp. 601-615

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality

||Id + T2||=1 + ||T2||

holds for every bounded linear operator T from X to X. This answers in the positive Question 4.11 of [Kadets, Martín, Merí; Norm equalities for operators, Indiana U. Math. J. 56 (2007), 2385--2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004), 151--183] and [Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004), 217--239]. We also construct compact spaces K1 and K2 such that C(K1) and C(K2) are extremely non-complex, C(K1) contains a complemented copy of C(2w) and C(K2) contains a (1-complemented) isometric copy of l-infinity.

  Piotr Koszmider; The interplay between compact spaces and the Banach spaces of   their continuous functions; in Open Problems in Topology 2; ed. Elliott Pearl,
  Elsevier 2007.

Many open problems in the isomorphic Banach space theory are related to the C(K)s and are left untouched sometimes for many decades. Some infinitary combinatorial ideas developed in the last three decades, so successful in the context of Ks, often have not yet been tested on the C(K)s. What follows aims at suggesting the tests.

  Piotr Koszmider; Kurepa trees and topological non-reflection; Topology and Its   Applications, vol 151, 2005, No. 1., pp. 77 - 98.

A property P of a structure S does not reflect if no substructure of S of smaller cardinality than S has the property. If for a given property P there is such an S of cardinality k, we say that P does not reflect at k. We undertake a fine analysis of Kurepa trees which results in defining canonical topological and combinatorial structures associated with the tree which possess a remarkably wide range of nonreflecting properties providing new constructions and solutions of open problems in topology. The most interesting results show that many known properties may not reflect at any fixed singular cardinal of uncountable cofinality. The topological properties we consider vary from normality, collectionwise Hausdorff property to metrizablity and many others. The combinatorial properties are related to stationary reflection.

   Piotr Koszmider; Projections in weakly compactly generated Banach spaces and
   Chang's Conjecture; Journal of Applied Analysis, 11 (2005), No. 2, 187--205
Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of identity (PRI) and the existence of many projections on separable subspaces (SCP). We address the questions if these can be the only projections in a nonseparable WCG space, in the sense that there is a PRI of projections P\alpha's for alpha between omega and lambda such that any projection is the sum of an operator in the closure of the linear span of countably many Palpha's (in the strong operator topology) and a separable range operator. Wark's modification of Shelah's and Steprans' construction provides an unconditional example for lambda equal to omega_1. We note that it is impossible for lambda biger than omega_2. The main result of the paper is that for lambda equal omega_2, the second uncountable cardinal, the question is logically undecidable and depends on additional axioms deciding the combinatorics on omega_2; for example Chang's conjecture implies that there are other projections than the projections mentioned above. The full strength results concern all linear operators not just the projections.

  Piotr Koszmider; A space C(K) where all non-trivial complemented subspaces have   big densities; Studia Mathematica 168 (2005), pp. 109 - 127
Using the method of forcing we prove that consistently there is a Banach space (of continuous functions on a zero-dimensional compact Hausdorff space) of density k bigger than the continuum where all operators are multiplications by a continuous function plus a weakly compact operator and which has no infinite dimensional complemented subspaces of density smaller or equal to the continuum. In particular no separable infinite dimensional subspace has a complemented superspace of density smaller or equal to the continuum, consistently answering a question of Johnson and Lindenstrauss of 1974.

  Piotr Koszmider; On Decompositions of Banach spaces of continuous functions on   Mrówka's spaces; Proceedings of the American Mathematical Society, 133 (2005),   pp. 2137 - 2146.
It is well-known that if K is infinite compact Hausdorff and scattered (i.e., with no perfect subsets) then the Banach space C(K) of continuous functions on K has complemented copies of c0. We address the question if it could be the only type of decompositions of C(K) not isomorfic to c0 into infinite dimensional summands for K infinite, scattered. Making a special set-theoretic assumption like the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question, providing apparently first examples of Banach spaces not isomorphic to c0 whose only non-trivial decompositions are into c0 and itself. The proofs use characterizations of operators as scalar multiples of the identity plus an operator with the range included in a copy of c0, i.e., our space has minimal possible space of operators among C(K)'s different than c0 for scattered K.

  Piotr Koszmider; Banach spaces of continuous functions with few operators;
  Matchematische Annalen, vol 330, (2004) No 1. pp 151 - 183
We present two constructions of infinite, separable, compact Hausdorff spaces K for which the Banach space C(K) of all continuous real-valued functions with the supremum norm has remarkable properties. In the first construction K is zero-dimensional and C(K) is non-isomorphic to any of its proper subspaces nor any of its proper quotients. In particular, it is an example of a C(K) space where the hyperplanes, one co-dimensional subspaces of C(K), are not isomorphic to C(K). In the second construction K is connected and C(K) is indecomposable which implies that it is not isomorphic to any C(K') for K's zero-dimensional. All these properties follow from the fact that there are few operators on our C(K)'s. If we assume the continuum hypothesis the spaces have few operators in the sense that every linear bounded operator T: from C(K) into C(K) is of the form gI+S where g is in C(K) and S is weakly compact or equivalently (in C(K) spaces) strictly singular.

   Marek Balcerzak, Artur Bartoszewicz, Piotr Koszmider; On Marczewski-Burstin   representable algebras; Colloquium Mathematicum, vol 99, No 1, 2004 pp. 55 - 60
We find first unconditional examples of algebras of sets which are not MB-representable. On the other hand, we prove that every Boolean algebra is isomorphic to an MB-representable algebra of sets.

  Piotr Koszmider; Universal matrices and strongly unbounded functions; Mathematical   Research Letters; Vol. 9, No 4. 2002, pp. 549 - 566
Fix an uncountable cardinal l. A symmetric matrix M=(mab)a,b < l whose entries are countable ordinals is called strongly universal if for every positive integer n, for every n×n matrix (bi j)i,j < n and for every uncountable set A={a: a Î A} Í [l]n of disjoint n-tuples a={a0,...an-1} there are a, a¢ Î A such that bij=mai aj¢ for 0 < i, j < n. We go beyond the recent dramatic discoveries for l = w1, w2 and address the question of the possibility of the existence of a strongly universal matrix for l > w2. Due to the undecidibility of some weak versions of the Ramsey property for l ³ w2 the positive answer can be at most consistent, but we show that well-investigated methods of forcing cannot yield that answer for l > w2. We use our method of "forcing with side conditions in semimorasses" to construct generically l by l strongly universal matrices for any cardinal l. The results are proved in more generality, related concepts are investigated, some questions are stated and some application are given.

  Piotr Koszmider, Artur Tomita, Steven Watson; Forcing countably compact group   topologies on a larger free Abelian groups; Topology Proceedings; Vol. 25. 2002
  pp. 563 - 574.
D.Dikranjan and D.Shakmatov asked for which cardinal k there exist a countably compact group topology on the free Abelian group of size k. M.Tkacenko obtained such group topology under the continuum hypothesis for the free Abelian group of size continuum. The second author showed that under MA, such group topology could be obtained for the free Abelian group of any size k equal to the continuum. Recently the second author and S.Watson improved those results to MA-countable. However it was still open whether a free Abelian group of size bigger than the continuum could be endowed with a countably compact group topology. The example below shows that this is true in a forcing model.

  Piotr Koszmider, Franklin D. Tall; A Lindelof space with no Lindelof subspace of size   aleph-one; Proceedings of the American Mathematical Society; Vol. 130 (2002)
  pp. 2777 - 2787.
A consistent example of an uncountable Lindelöf T3 (and hence normal) space with no Lindelöf subspace of size aleph-one is constructed. It remains unsolved whether extra set-theoretic assumptions are necessary for the existence of such a space. However, our space has size aleph-two and is a P-space, i.e., Gd's are open, and for such spaces extra set-theoretic assumptions turn out to be necessary.

   Lúcia Junqueira, Piotr Koszmider; On families of Lindelöf and related subspaces of two to omega one.    Fundamenta Matematicae; Vol. 169 (2001), no. 3, 205-231.
"We consider the families of all subspaces of size omega-one of two-to-omega-one (or of a compact zero dimensional space X of weight omega-one in general) which are normal, have Lindelof property or are closed under limits of convergent omega-one-sequences. Various relations among these families modulo the club filter subsets of X are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X_M for an elementary submodel M of size omega-one. Various results with this flavor are obtained. The other tool used is forcing and in this case various preservation or non-preservation results of topological and combinatorial properties are proved. In particular we prove that there may be no c.c.c. forcing which destroys Lindelof property of compact spaces, answering a question of Juhász. Many related questions are formulated.

    Piotr Koszmider; On Strong Chains of uncountable functions; Israel Journal of
    Mathematics; 2000, Vol 118, 289 - 315.
For functions f,g:w1®w1, where w1 is the first uncountable cardinal, we write that f << g if and only if {x Î w1: f(x) is less or equal g(x)} is finite. We prove the consistency of the existence of a well-ordered increasing << -chain of length w2, solving a problem of A. Hajnal. The methods involve previously developed by us forcing with side conditions in morasses which is a variation on Todorcevic's forcing with models as side conditions. The paper is self contained and requires from the reader the knowledge of Kunen's textbook and some basic experience with proper forcing and elementary submodels.

  Piotr Koszmider; Forcing minimal extensions of Boolean algebras; 1999, Vol. 351,   3073-3117.
We employ a forcing approach to extending a Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications:
1) We make Fedorchuk's method more flexible, obtaining for every cardinal l of uncountable cofinality, a consistent example of a Boolean algebra Al whose every infinite homomorphic image is of cardinality l and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight l). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra A can be strictly less that the minimal size of a homomorphic image of A, answering a question of J. D. Monk.
2) We prove that for every cardinal of uncountable cofinality it is consistent that 2w=l and both Al and Aw1 exist.
3) By combining these algebras we obtain many examples that answer questions of J.D. Monk.
4) We prove the consistency of MA + not CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangel'skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character.
5) We prove that for any cardinal l of uncountable cofinality it is consistent that there is a countably tight Boolean algebra A with a distinguished ultrafilter ¥ such that for every a\not ' ¥ the algebra A|a is countable and ¥ has hereditary character l.

  Piotr Koszmider; On the existenceof strong chains in P(w_1)/Fin. Journal of Symbolic   Logic 1998, vol. 63. no. 3.
(Xa:a in w2) included in P(w1) is a strong chain in P(w1)/Fin if and only if Xb-Xa is finite and Xa-Xb is uncountable for each b < a < w1. We show that it is consistent that a strong chain in P(w1) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in P(w1) but no strong chain exists: box-w1 is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain. We also show that forcing a strong chain in w1w1/Fin would require another method.

  Piotr Koszmider; Applications of Rho-functions; In Set Theory: Techniques and   Applications (eds. C. Di Prisco, J. Larson, J. Bagaria, A. Mathias) Kluwer Acad.   Publishers, Dotrecht, 1998, 83-98.
"We will provide below examples of these canonical methods and by this we would like to defend the statement that" rho-functions are the most important functions of two-cardinal combinatorics"

  Piotr Koszmider; Models as side conditions; In Set Theory: Techniques and   Applications (eds. C. Di Prisco, J. Larson, J. Bagaria, A. Mathias) Kluwer Acad.   Publishers, Dotrecht, 1998, 99-107.

"The main idea of the method of models as side conditions is to construct a forcing notion whose conditions p are of the form p=(D,N), where D is the working part and comes from the original forcing with finite conditions and N is an elementary chain of countable elementary submodels."

  Gary Gruenhage, Piotr Koszmider; Arhangel'skii-Tall problem: A consistent   counterexample; Fundamenta Matematicae; 1996, Vol 149, 275-285.

We construct a consistent examples of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arhangelskii and F. Tall. An interplay between a tower in P(w)/Fin, an almost disjoint family in [w]^w, and a version of an (w,1)-morass forms the core of the proof. A part of the poset which forces the couterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

  Gary Gruenhage, Piotr Koszmider; Arhangel'skii-Tall problem under Martin's Axiom;   Fundamenta Matematicae; 1995, Vol 149, 143-166.
We show that under MA(sigma-centred)(omega-one) implies that normal locally compact metacompact spaces are paracompact, and that MA(omega_1) implies normal locally compact metalindelof spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is normal locally compact metalindelof space which is not paracompact

   Katsuya Eda, Gary Gruenhage, Piotr Koszmider, Kenichi Tamano,
   Stevo Todorcevic; Sequential Fans in Topology; Topology and Its Applications,   1995, Vol 67, no 3, 189-220.
For an index set I, let S(I) be the sequential fan with I spines, i.e., the topological sum of I copies of the convergent sequence with all nonisolated points identified. The simplicity and the combinatorial nature of this space is what lies behind its occurrences in many seemingly unrelated topological problems.

  Piotr Koszmider; Semimorasses and Nonreflection at Singular Cardinals; Annals of   Pure and Applied Logic; 1995, Vol 72, 1-23.
Some subfamilies of P k(l), for k regular, k <l, called (k,l)-semimorasses are investigated. For l = k+, they constitute weak versions of Velleman's simplified (k,1)-morasses, and for l > k+, they provide a combinatorial framework which in some cases has similar applications to the application of (k,1)-morasses with this difference that the obtained objects are of size > k+, and not only of size k+ as in the case of morasses. New consistency results involve (compatible with CH) existence of nonreflecting objects of singular sizes of uncountable cofinality such as a nonreflecting stationary set in P k(l), a nonreflecting nonmetrizable space of size l, a nonreflecting nonspecial tree of size l. We also characterize possible minimal sizes of nonspecial trees without uncountable branches.

  Piotr Koszmider; On Coherent Families of Finite-to-One Functions; Journal of   Symbolic Logic, 1993, Vol. 58 (1), 128-138.
We consider the existence of coherent families of finite-to-one functions on countable subsets of an uncountable cardinal kappa. The existence of such families for kappa implies the existence of a winning 2-tactic for player TWO in the countable-finite game on kappa answering a question of M. Scheepers. We prove that coherent families exist on kappa=omega_n, where n\in\omega, and that they consistently exist for every cardinal kappa. We also prove that iterations of Axiom A forcings with countable supports are Axiom A.

  Piotr Koszmider; On The Complete Invariance Property in Some Uncountable   Products; Canadian Mathematical Bulletin ; 1992, Vol. 35 (2), 221-229.
We consider uncountable products of nontrivial compact, convex subsets of Banach spaces. We show that these products do not have the complete invariance property i.e. they include a nonempty, closed subset which is not a fixed point set (i.e. the set of all fixed points) for any continuous mapping from the product into itself. In particular we give an answer to W.Weiss' question whether uncountable powers of the unit interval have the complete invariance property.

   Piotr Koszmider; A Formalism for Some Class of Forcing Notions; Zeitschrift fur
  Mathematische Logic und Grundlagen der Mathematik, 1992, Vol. 38, 413-421.
We introduce a class of forcing notions of type S, which contains among others, Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for these forcing notions, which allows all standard trics for iterations or products with countable supports of Sacks forcing. On the other hand, it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports.

   Winfried Just, Piotr Koszmider; Remarks on Cofinalities and Homomorphism Types   of Boolean Algebras; Algebra Universalis 1991, Vol. 28, 138-149.
We show that it is consistent that there are Boolean algebras of cardinalities smaller than the continuum which have no infinite countable homomorphic images, which by the Stone duality means that there can are compact Hausdorff spaces with no convergent sequences of weights less than the continuum (unlike the classical compact spaec with no converging sequences the beta N). We also show that there are consistent examples of Boolean algebras of cardinalities smaller than the continuum whose cofinality is uncountable, showing that the assumption of Martin's axiom is necessary in a known result of S. Koppeleberg.

  Piotr Koszmider; The Consistency of the negation of CH and the pseudoaltitude less   or equal to omega one; Algebra Universalis 1990, Vol. 27, 80-87.
We show that it is consistent that the negation of the continuum hypothesis holds and all Boolean algebras have pseudoaltitude less or equal to omega-one. It follows that the following dichotomy is consistent: Either a Boolean algebra has an uncountable independent family or it has an infinite homomorphic image with an ultrafilter of character less or equal to omega_1. Another topological version of a conlusion is: Either a compact separable Hausdorff space maps onto a nonmetrizable Cantor cube or has a nonisolated point of small character (omega or omega_1) in the presence of arbitrarily large continuum.