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Tomasz Kochanek

Publications and source records attributed to Tomasz Kochanek.

At least 19 recordsLinked to original sources
arXiv2024

When is the Szlenk derivation of a dual unit ball another ball?

We show that if a separable Banach space has Kalton's property $(M^\ast)$, then all $\varepsilon$-Szlenk derivations of the dual unit ball are balls, however, in the case of the dual of Baernstein's space, all those Szlenk derivations are balls having the same radius as for $\ell_2$, yet this space fails property $(M^\ast)$. By estimating the radii of enveloping balls, we show that the Szlenk derivations are not balls for Tsirelson's space and the dual of Schlumprecht's space. Using the Karush-Kuhn-Tucker theorem we prove that the same is true for the duals of certain sequential Orlicz spaces.

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arXiv2024

Enveloping balls of Szlenk derivations

For Banach spaces with a shrinking FDD, we provide estimates for the radii of the enveloping balls of the $\varepsilon$-Szlenk derivations of the dual unit ball.

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arXiv2024

A note on ideal C$^\ast$-completions and amenability

For a discrete group $G$, we consider certain ideals $\mathcal{I}\subset c_0(G)$ of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C$^\ast$-algebra of $G$ and the C$^\ast$-completion $\mathrm{C}_{\mathcal{I}}^\ast(G)$ in the sense of Brown and Guentner implies that $G$ is amenable.

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arXiv2024

Automatic continuity of operator semigroups in the Calkin algebra

We study operator semigroups in the Calkin algebra $\mathcal{Q}(\mathcal{H})$, represented as a subalgebra of the algebra of bounded linear operators on a Hilbert space via one of `canonical' Calkin's representations. Using the BDF theory, we associate with any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathcal{Q}(\mathcal{H})$ an extension $Γ\in\mathrm{Ext}(Δ)$, where $Δ$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $σ(A)$ of the generator of $(q(t))_{t\geq 0}$. Then we show that, in natural circumstances, if $(q(t))_{t\geq 0}$ is continuous in the strong operator topology, then it is actually uniformly continuous, although there are $C_0$-semigroups in $\mathcal{Q}(\mathcal{H})$ that are not uniformly continuous.

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arXiv2023

Compact perturbations of operator semigroups

We study lifting problems for operator semigroups in the Calkin algebra $\mathscr{Q}(\mathcal{H})$, our approach being mainly based on the Brown--Douglas--Fillmore theory. With any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathscr{Q}(\mathcal{H})$ we associate an extension $Γ\in\mathrm{Ext}(Δ)$, where $Δ$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $σ(A)$ of the generator of $(q(t))_{t\geq 0}$. By using Milnor's exact sequence, we show that if each $q(t)$ has a normal lift, then the question whether $Γ$ is trivial reduces to the question whether the corresponding first derived functor vanishes. With the aid of the CRISP property and Kasparov's Technical Theorem, we provide geometric conditions on $σ(A)$ which guarantee splitting of $Γ$. If $Δ$ is a perfect compact metric space, we obtain in this way a $C_0$-semigroup $(Q(t))_{t\geq 0}$ which lifts $(q(t))_{t\geq 0}$ on dyadic rationals.

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arXiv2019

Approximately order zero maps between C*-algebras

We investigate linear operators between C$^\ast$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\varepsilon>0$ we have $\|ϕ(x)ϕ(y)\|\leq\varepsilon\|x\|\|y\|$ for all positive $x,y$ with $xy=0$. We establish some structural properties of such maps concerning approximate Jordan-like equations and almost commutation relations. In some situations (e.g. when the codomain is finite-dimensional), we show that $ϕ$ can be approximated by an approximate Jordan $^\ast$-homomorphism, with both errors depending only on $\|ϕ\|$ and $\varepsilon$.

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arXiv2018

Operator ideals and three-space properties of asymptotic ideal seminorms

We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the `automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk power type $\max\{p,q\}$.

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arXiv2017

Approximate homomorphisms on lattices

We prove two results concerning an Ulam-type stability problem for homomorphisms between lattices. One of them involves estimates by quite general error functions; the other deals with approximate (join) homomorphisms in terms of certain systems of lattice neighborhoods. As a corollary, we obtain a stability result for approximately monotone functions.

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arXiv2016

Steinhaus' lattice-point problem for Banach spaces

Steinhaus proved that given a~positive integer $n$, one may find a circle surrounding exactly $n$ points of the integer lattice. This statement has been recently extended to Hilbert spaces by Zwoleński, who replaced the integer lattice by any infinite set that intersects every ball in at most finitely many points. We investigate Banach spaces satisfying this property, which we call (S), and characterise them by means of a new geometric property of the unit sphere which allows us to show, e.g., that all strictly convex norms have (S), nonetheless, there are plenty of non-strictly convex norms satisfying (S). We also study the corresponding renorming problem.

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arXiv2016

The Szlenk power type and tensor products of Banach spaces

We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $ω$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.

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arXiv2016

Uncountable sets of unit vectors that are separated by more than 1

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ that depends only on $X$. If $K$ is a non-metrisable compact, Hausdorff space, then the unit sphere of $X=C(K)$ also contains such a subset; if moreover $K$ is perfectly normal, then one can find such a set with cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.

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arXiv2015

Direct sums and summability of the Szlenk index

We prove that the $c_0$-sum of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums. We also give a formula for the Szlenk power type of the $\mathfrak{E}$-direct sum of separable spaces provided that $\mathfrak{E}$ has a shrinking unconditional basis whose dual basis yields an asymptotic $\ell_p$ structure in $\mathfrak{E}^\ast$. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of $c_0$ has power type $1$ but non-summable Szlenk index.

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arXiv2013

The ideal of weakly compactly generated operators acting on a Banach space

We call a bounded linear operator acting between Banach spaces weakly compactly generated ($\mathsf{WCG}$ for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of $\mathsf{WCG}$ operators, we prove that it forms a closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the $p$th long James space $\mathcal{J}_p(ω_1)$, we show how properties of the ideal of $\mathsf{WCG}$ operators (such as being the unique maximal ideal) may be used to derive results outside ideal theory. For instance, we identify the $K_0$-group of $\mathscr{B}(\mathcal{J}_p(ω_1))$ as the additive group of integers.

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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.

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arXiv2013

Stability of vector measures and twisted sums of Banach spaces

A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector measures) property if there exists a constant $v<\infty$ such that for any algebra of sets $\mathcal F$, and any function $ν\colon\mathcal F\to X$ satisfying $$\|ν(A\cup B)-ν(A)-ν(B)\|\leq 1\quad{for disjoint}A,B\in\mathcal F,$$there is a vector measure $μ\colon\mathcal F\to X$ with $\|ν(A)-μ(A)\|\leq v$ for all $A\in\mathcal F$. If this condition is valid when restricted to set algebras $\mathcal F$ of cardinality less than some fixed cardinal number $κ$, then we say that $X$ has the $κ$-$\mathsf{SVM}$ property. The least cardinal $κ$ for which $X$ does not have the $κ$-$\mathsf{SVM}$ property (if it exists) is called the $\mathsf{SVM}$ character of $X$. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine $\mathsf{SVM}$ characters for many classical Banach spaces. We also discuss connections between the $κ$-$\mathsf{SVM}$ property, $κ$-injectivity and the `three-space' problem.

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arXiv2013

A chain condition for operators from C(K)-spaces

We introduce a chain condition (bishop), defined for operators acting on C(K)-spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pełczyński's characterisation of weakly compact operators on C(K)-spaces. We prove that if K is extremally disconnected and X is a Banach space then an operator T : C(K) -> X is weakly compact if and only if it satisfies (bishop) if and only if the representing vector measure of T satisfies an analogous chain condition. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal's lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies (bishop), for example both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying (bishop) forms a closed left ideal of B(C(K)).

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