arXiv subjects

Piotr Koszmider

Publications and source records attributed to Piotr Koszmider.

At least 37 records · Page 2Linked to original sources

arXiv2019

On $\mathbb R$-embeddability of almost disjoint families and Akemann-Doner C*-algebras

An almost disjoint family $\mathcal A$ of subsets of $\mathbb N$ is said to be $\mathbb R$-embeddable if there is a function $f:\mathbb N\rightarrow \mathbb R$ such that the sets $f[A]$ are ranges of real sequences converging to distinct reals for distinct $A\in \mathcal A$. It is well known that almost disjoint families which have few separations, such as Luzin families, are not $\mathbb R$-embeddable. We study extraction principles related to $\mathbb R$-embeddability and separation properties of almost disjoint families of $\mathbb N$ as well as their limitations. An extraction principle whose consistency is our main result is: every almost disjoint family of size continuum contains an $\mathbb R$-embeddable subfamily of size continuum. It is true in the Sacks model. The Cohen model serves to show that the above principle does not follow from the fact that every almost disjoint family of size continuum has two separated subfamilies of size continuum. We also construct in ZFC an almost disjoint family, where no two uncountable subfamilies can be separated but always a countable subfamily can be separated from any disjoint subfamily. Using a refinement of the $\mathbb R$-embeddability property called a controlled $\mathbb R$-embedding property we obtain the following results concerning Akemann-Doner C*-algebras which are induced by uncountable almost disjoint families: a) In ZFC there are Akemann-Doner C*-algebras of density $\mathfrak c$ with no commutative subalgebras of density $\mathfrak c$, b) It is independent from ZFC whether there is an Akemann-Doner algebra of density $\mathfrak c$ with no nonseparable commutative subalgebra. This completes an earlier result that there is in ZFC an Akemann-Doner algebra of density $ω_1$ with no nonseparable commutative subalgebra.

math.LO↗

arXiv2018

An extension of compact operators by compact operators with no nontrivial multipliers

We construct an essential extension of $\mathcal K(\ell_2({\mathfrak{c}}))$ by $\mathcal K(\ell_2)$, where ${\mathfrak{c}}$ denotes the cardinality of continuum, i.e., a $C^*$-algebra $\mathcal A\subseteq \mathcal B(\ell_2)$ satisfying the short exact sequence $$0\rightarrow \mathcal K(\ell_2)\xrightarrowι \mathcal A \rightarrow\mathcal K(\ell_2({\mathfrak{c}}))\rightarrow 0,$$ where $ι[\mathcal K(\ell_2)]$ is an essential ideal of $\mathcal A$ such that the algebra of multipliers $\mathcal M(\mathcal A)$ of $\mathcal A$ is equal to the unitization of $\mathcal A$. In particular $\mathcal A$ 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 $\mathcal K(\ell_2)$, 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 ${\mathbb{N}}$. The role of the almost disjoint family is played by an almost orthogonal family of projections in $\mathcal B(\ell_2)$, but the almost matrix units corresponding to the matrix units in $\mathcal K(\ell_2({\mathfrak{c}}))$ must be constructed with extra care. This example may also contribute in the future to our understanding of the semigroups $Ext(\mathcal K(\ell_2(κ)))$ for $ω_1\leq κ\leq\mathfrak{c}$ which are unexplored at the moment.

math.OA↗

arXiv2018

Large irredundant sets in operator algebras

A subset $\mathcal X$ of a C*-algebra $\mathcal A$ is called irredundant if no $A\in \mathcal X$ belongs to the C*-subalgebra of $\mathcal A$ generated by $\mathcal X\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum. There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms and we investigate here the noncommutative case. Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis) we prove that there is an AF C*-subalgebra of $\mathcal B(\ell_2)$ of density $2^ω=ω_1$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in $\mathcal B(\ell_2)$ of cardinality continuum contains an irredundant subcollection of cardinality continuum. Other partial results and more open problems are presented.

math.OA↗

arXiv2018

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 $\mathcal A$, using the ideal $\mathcal I^{At}(\mathcal A)$ generated by the minimal projections of $\mathcal A$. 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 $\mathcal A_α$, along an uncountable limit ordinal $λ$, such that $\mathcal A_{α+1}/\mathcal A_α$ is $*$-isomorphic to the algebra of all compact operators on a separable Hilbert space and $\mathcal A_{α+1}$ is $σ$-unital and stable for each $α<λ$, but $\mathcal A$ is not stable and where all ideals of $\mathcal A$ are of the form $\mathcal A_α$. In particular, $\mathcal A$ is a nonseparable $C^*$-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. R\ordam 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 finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers.

math.OA↗

arXiv2018

C*-Algebras With and Without $\ll$-Increasing Approximate Units

For elements $a, b$ of a C*-algebra we denote $a=ab$ by $a\ll b$. We show that all $ω_1$-unital C*-algebras have $\ll$-increasing approximate units, extending a classical result for $σ$-unital C*-algebras. We also construct (in ZFC) the first examples of C*-algebras with no $\ll$-increasing approximate unit. One of these examples is a C*-subalgebra of $\mathcal B(\ell_2(ω_1))$. These examples are, by necessity, not approximately finite dimensional (AF), but some of them 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. We further show that the existence of a C*-subalgebra of $\mathcal B(\ell_2)$ with no $\ll$-increasing approximate unit or the existence of an LF but not AF C*-subalgebra of $\mathcal B(\ell_2)$ are independent of ZFC. Our examples also show that an extension of an AF algebra by an AF algebra does not have to be an AF algebra in the nonseparable case.

math.OA↗

arXiv2017

A non-stable C*-algebra with an elementary essential composition series

A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C*-algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra $A$ of $B(\ell_2)$, which is the inductive limit of length $ω_1$ of its separable stable ideals $I_α$ ($α<ω_1$) satisfying $I_{α+1}/I_α\cong K(\ell_2)$ for each $α<ω_1$, while $A$ is not stable. The sequence $(I_α)_{α\leqω_1}$ is the GCR composition series of $A$ which in this case coincides with the Cantor-Bendixson composition series as a scattered C*-algebra. $A$ has the property that all of its proper two-sided ideals are listed as $I_α$s for some $α<ω_1$ and therefore the family of stable ideals of $A$ has no maximal element. By taking $A'=A\otimes K(\ell_2)$ we obtain a stable C*-algebra with analogous composition series $(J_α)_{α<ω_1}$ whose ideals $J_α$s are isomorphic to $I_α$s for each $α<ω_1$. In particular, there are nonisomorphic scattered C*-algebras whose GCR composition series $(I_α)_{α\leqω_1}$ satisfy $I_{α+1}/I_α\cong K(\ell_2)$ for all $α<ω_1$, for which the composition series differ first at $α=ω_1$.

math.OA↗

arXiv2017

A 1-separably injective space that does not contain $\ell_\infty$

We show that the problem whether every $1$-separably injective Banach space contains an isomorphic copy of $\ell_\infty$ is undecidable. Namely, unlike under the continuum hypothesis, assuming Martin's axiom and the negation of the continuum hypothesis, there is an $1$-separably injective Banach space of the form $C(K)$ (which means that $K$ is an $F$-space) without an isomorphic copy of $\ell_\infty$. This result is a consequence of our study of $ω_2$-subsets of tightly $σ$-filtered Boolean algebras introduced by Koppelberg for which we obtain some general principles useful when transferring properties of Boolean algebras to the level of Banach spaces.

math.FA↗

arXiv2017

On constructions with $2$-cardinals

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 $f:[λ^{++}]^2\rightarrow[λ^{++}]^{\leqλ}$ with the appropriate $λ^+$-version of property $Δ$ for regular $λ\geqω$ satisfying $λ^{<λ}=λ$.

math.LO↗

arXiv2016

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 $X\oplus Y$, 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 $\ell_\infty$ 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 $g\in C(K)$ 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.

math.FA↗

arXiv2016

A note on the Akemann-Doner and Farah-Wofsey constructions

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 $\aleph_1$ 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.

math.OA↗

arXiv2015

On the problem of compact totally disconnected reflection of nonmetrizability

We construct a ZFC example of a nonmetrizable compact space $K$ such that every totally disconnected closed subspace $L\subseteq K$ 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 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.

math.GN↗

arXiv2015

An isometrically universal Banach space induced by a non-universal Boolean algebra

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 $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ 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 $κ$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $κ$. 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).

math.FA↗

arXiv2015

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 $(1+\varepsilon)$-separated set in the unit sphere for any $\varepsilon>0$) 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.

math.FA↗

arXiv2015

On complemented copies of $c_0(ω_1)$ in $C(K^n)$ spaces

Given a compact Hausdorff space $K$ we consider the Banach space of real continuous functions $C(K^n)$ or equivalently the $n$-fold injective tensor product $\hat\bigotimes_{\varepsilon}C(K)$ 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 $c_0(ω_1)$ in $\hat\bigotimes_{\varepsilon}C(K)$ under the hypothesis that $C(K)$ contains an isomorphic copy of $c_0(ω_1)$. This is related to the results of E. Saab and P. Saab that $X\hat\otimes_\varepsilon Y$ contains a complemented copy of $c_0$, if one of the infinite dimensional Banach spaces $X$ or $Y$ contains a copy of $c_0$ and of E. M. Galego and J. Hagler that it follows from Martin's Maximum that if $C(K)$ has density $ω_1$ and contains a copy of $c_0(ω_1)$, then $C(K\times K)$ contains a complemented copy $c_0(ω_1)$. The main result is that under the assumption of $\clubsuit$ for every $n\in N$ there is a compact Hausdorff space $K_n$ of weight $ω_1$ such that $C(K)$ is Lindelöf in the weak topology, $C(K_n)$ contains a copy of $c_0(ω_1)$, $C(K_n^n)$ does not contain a complemented copy of $c_0(ω_1)$ while $C(K_n^{n+1})$ does contain a complemented copy of $c_0(ω_1)$. 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.

math.FA↗

arXiv2015

On automorphisms of the Banach space $\ell_\infty/c_0$

We investigate Banach space automorphisms $T:\ell_\infty/c_0\rightarrow\ell_\infty/c_0 $ focusing on the possibility of representing their fragments of the form $$T_{B,A}:\ell_\infty(A)/c_0(A)\rightarrow \ell_\infty(B)/c_0(B)$$ for $A, B\subseteq N$ infinite by means of linear operators from $\ell_\infty(A)$ into $\ell_\infty(B)$, infinite $A\times B$-matrices, continuous maps from $B^*=βB\setminus B$ into $A^*$, or bijections from $B$ to $A$. This leads to the analysis of general linear operators on $\ell_\infty/c_0$. 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 $\ell_\infty/c_0$ which cannot be lifted to operators on $\ell_\infty$ and assuming OCA+MA we show that every automorphism of $\ell_\infty/c_0$ with no fountains or with no funnels is locally, i.e., for some infinite $A, B\subseteq N$ 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.

math.FA↗

arXiv2014

A weak*-topological dichotomy with applications in operator theory

Denote by $[0,ω_1)$ the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let $C_0[0,ω_1)$ be the Banach space of scalar-valued, continuous functions which are defined on $[0,ω_1)$ and vanish eventually. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,ω_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,ω_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,ω_1)$ can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of $C_0[0,ω_1)$ 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 $C_0[0,ω_1)$. As a consequence, we find that this ideal has a bounded left approximate identity, thus resolving 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 $C_0[0,ω_1)$ and the operators acting on it.

math.FA↗

arXiv2014

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 $\mathscr{B}(X)$ of all bounded operators on $X$, and suppose that $Y$ contains a complemented subspace which is isomorphic to $Y\oplus Y$ and that $X$ is isomorphic to $X\oplus Z$ for every complemented subspace $Z$ of $Y$. Then the $K_0$-group of $\mathscr{B}(X)$ is isomorphic to the additive group $\mathbb{Z}$ of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that $K_0(\mathscr{B}(C([0,ω_1])))\cong\mathbb{Z}$, where $C([0,ω_1])$ denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal $ω_1$, endowed with the order topology.

math.KT↗

arXiv2013

Maximal left ideals of the Banach algebra of bounded operators on a Banach space

We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ 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: (i) Question (Q1) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (Q2) has a positive answer if and only if no finitely-generated, maximal left ideal of $\mathscr{B}(E)$ contains $\mathscr{F}(E)$; (iii) the answer to Question (Q2) is positive for many, but not all, Banach spaces.

math.FA↗