Preprints

1. Approximately order zero maps between C*-algebras arXiv: 1911.01689v1
Abstract
We investigate linear operators between C∗-algebras which approximately preserve involution and orthogonality, the latter meaning that for some \(\varepsilon>0\) we have \(\|\phi(x)\phi(y)\|\leq\varepsilon\|x\|\|y\|\) 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 \(\phi\) can be approximated by an approximate Jordan ^∗-homomorphism, with both errors depending only on \(\|\phi\|\) and \(\varepsilon\).

Published papers

1. Operator ideals and three-space properties of asymptotic ideal seminorms
   Transactions of the American Mathematical Society 371 (2019), 8173-8215 arXiv
   (jointly with R.M. Causey and S. Draga)
Abstract
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 \(\max\{p,q\}\).
2. Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete
   Bulletin of the London Mathematical Society 50 (2018), 680-696 arXiv
   (jointly with E. Pernecká)
Abstract
Let \(M\) be a compact subset of a superreflexive Banach space. We prove that the Lipschitz-free space \(\mathcal{F}(M)\), the predual of the Banach space of Lipschitz functions on \(M\), has the Pełczyński's property (V\(^\ast\)). As a consequence, the Lipschitz-free space \(\mathcal{F}(M)\) is weakly sequentially complete.
3. Approximate homomorphisms on lattices
   Archiv der Mathematik (Basel) 111 (2018), 177-186 arXiv
   (jointly with R. Badora and B. Przebieracz)
Abstract
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.
4. The Szlenk power type and tensor products of Banach spaces
   Proceedings of the American Mathematical Society 145 (2017), 1685-1698 ResearchGate
   (jointly with S. Draga)
Abstract
We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most \(\omega\). 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.
5. Steinhaus' lattice-point problem for Banach spaces
   Journal of Mathematical Analysis and Applications 446 (2017), 1219-1229 arXiv
   (jointly with T. Kania)
Abstract
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.
6. Direct sums and summability of the Szlenk index
   Journal of Functional Analysis 271 (2016), 642-671 ResearchGate
   (jointly with S. Draga)
Abstract
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 \(E\)-direct sum of separable spaces provided that \(E\) has a shrinking unconditional basis whose dual basis yields an asymptotic \(\ell_p\) structure in \(E^\ast\). 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.
7. Uncountable sets of unit vectors that are separated by more than 1
   Studia Mathematica 232 (2016), 19-44 arXiv
   (jointly with T. Kania)
Abstract
Let \(X\) be a Banach space. We study the circumstances under which there exists an uncountable set \(A\subset X\) of unit vectors such that \(\|x-y\|>1\) for distinct \(x,y\in A\). 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 \(\|x-y\|\geq 1+\varepsilon\) 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.
8. A chain condition for operators from C(K)-spaces
   Quarterly Journal of Mathematics 65 (2014), 703-715 arXiv
   (jointly with K.P. Hart and T. Kania)
Abstract
We introduce a chain condition (), 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 \(T\colon C(K)\to X\) is weakly compact if and only if it satisfies () 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 (), 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 () forms a closed left ideal of \(\mathscr{B}(C(K))\).
9. The ideal of weakly compactly generated operators acting on a Banach space
   Journal of Operator Theory 71 (2014), 455-477 arXiv
   (jointly with T. Kania)
Abstract
We call a bounded linear operator acting between Banach spaces weakly compactly generated (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 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 \(\mathcal{J}_p(\omega_1)\), we show how properties of the ideal of 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 \(\mathscr{B}(\mathcal{J}_p(\omega_1))\) as the additive group of integers.
10. Stability of vector measures and twisted sums of Banach spaces
    Journal of Functional Analysis 264 (2013), 2416-2456 arXiv
Abstract
A Banach space \(X\) is said to have the 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 \(\nu\colon\mathcal{F}\to X\) satisfying $$\|\nu (A\cup B)-\nu(A)-\nu(B)\|\leq 1 \quad\mbox{for disjoint }A,\, B\in\mathcal{F}$$ there is a vector measure \(\mu\colon\mathcal{F}\to X\) with \(\|\nu(A)-\mu(A)\|\leq v\) 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 \(\kappa\), then we say that \(X\) has the \(\kappa\)-SVM property. The least cardinal \(\kappa\) for which \(X\) does not have the \(\kappa\)-SVM property (if it exists) is called the SVM character of \(X\). We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine SVM characters for many classical Banach spaces. We also discuss connections between the \(\kappa\)-SVM property, \(\kappa\)-injectivity and the `three-space' problem.
11. Maximal left ideals of the Banach algebra of bounded operators on a Banach space
    Studia Mathematica 218 (2013), 245-286 arXiv
    (jointly with H.G. Dales, T. Kania, P. Koszmider and N.J. Laustsen)
Abstract
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 space \(E\):
(I) Does \(\mathscr{B}(E)\) always contain a maximal left ideal which is not finitely generated?
(II) Is every finitely-generated, maximal left ideal of \(\mathscr{B}(E)\) necessarily of the form $$(\ast)\qquad\{T\in\mathcal{B}(E)\colon\, Tx=0\}\quad$$ 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 (\(\ast\)), a positive answer to the second question would imply a positive answer to the first. Our main results are: (i) Question (I) has a positive answer for most (possibly all) infinite-dimensional Banach spaces; (ii) Question (II) 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 (II) is positive for many, but not all, Banach spaces.
12. Characterisation of L_p-norms via Hölder's inequality
    Journal of Mathematical Analysis and Applications 399 (2013), 403-410 ResearchGate
    (jointly with M. Lewicki)
Abstract
We characterise \(L_p\)-norms on the space of integrable step functions, defined on a probabilistic space, via Hölder's type inequality with an optimality condition.
13. Almost orthogonally additive functions
    Journal of Mathematical Analysis and Applications 400 (2013), 1-14 ResearchGate
    (jointly with W. Wyrobek-Kochanek)
Abstract
If a function \(f\), acting on a Euclidean space \(\mathbb{R}^n\), is "almost" orthogonally additive in the sense that \(f(x+y)=f(x)+f(y)\) for all \((x,y)\in\bot\setminus Z\), where \(Z\) is a "negligible" subset of the \((2n-1)\)-dimensional manifold \(\bot\subset\mathbb{R}^{2n}\), then \(f\) coincides almost everywhere with some orthogonally additive mapping.
14. \(\mathcal{F}\)-bases with brackets and with individual brackets in Banach spaces
    Studia Mathematica 211 (2012), 259-268 arXiv
Abstract
We provide a partial answer to the question of Vladimir Kadets whether given an \(\mathcal{F}\)-basis of a Banach space \(X\), with respect to some filter \(\mathcal{F}\subset\mathcal{P}(\mathbb{N})\), the coordinate functionals are continuous. The answer is positive if the character of \(\mathcal{F}\) is less than \(\mathfrak{p}\). In this case every \(\mathcal{F}\)-basis is an \(M\)-basis with brackets which are determined by an element of \(\mathcal{F}\).
15. On measurable solutions of a general functional equation on topological groups
    Publicationes Mathematicae Debrecen 78 (2011), 527-533
    (jointly with M. Lewicki)
Abstract
We establish a theorem of the type "measurability implies continuity" for solutions \(f\) of the functional equation $$\Gamma(f(x),f(y))=\Phi(x,y,f(\alpha_1x+\beta_1y),\ldots ,f(\alpha_nx+\beta_ny))$$ under reasonable conditions upon the integers \(\alpha_i,\beta_i\) and the mappings \(\Gamma,\Phi\).
16. Corrigendum to "Stability aspects of arithmetic functions II" (Acta Arith. 139 (2009), 131-146)
    Acta Arithmetica 149 (2011), 83-98 ResearchGate
Abstract
The main results from the paper mentioned in the title are corrected. For instance, in the additive case the following theorem is proved: Let \(f\colon\mathbb{N}\to\mathbb{R}\) be an arithmetic function satisfying $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\vert f(xy)-f(x)-f(y)\vert\leq\varepsilon$$ and $$x,y\in\mathbb{N},\,\{p\in\mathbb{P}\colon p\mid x\}=\{p\in\mathbb{P}\colon p\mid y\}\,\,\Rightarrow\,\,\vert f(x)-f(y)\vert\leq 2\varepsilon.$$ Then there exists a real strongly additive function \(\widetilde{f}\) such that \(\vert f(x)-\widetilde{f}(x)\vert\leq K^\ast\varepsilon\) for all \(x\in\mathbb{N}\) with some absolute constant \(K^\ast\leq 89/2\).
17. On a composite functional equation fulfilled by modulus of an additive function
    Aequationes Mathematicae 80 (2010), 155-172 ResearchGate
Abstract
We deal with the problem of determining general solutions \(f\colon\mathbb{R}\to\mathbb{R}\) of the following composite functional equation introduced by Fechner: $$f(f(x)-f(y))=f(x+y)+f(x-y)-f(x)-f(y).$$ Our result gives a partial answer to this problem under some assumptions upon \(f(\mathbb{R})\). We are applying a theorem of Simon and Volkmann concerning a certain characterization of modulus of an additive function. A new proof of their result is also presented.
18. Probability distribution solutions of a general linear equation of infinite order II
    Annales Polonici Mathematici 99 (2010), 215-224
    (jointly with J. Morawiec)
Abstract
Let \((\Omega,\mathcal{A},P)\) be a probability space and let \(\tau\colon\mathbb{R}\times\Omega\to\mathbb{R}\) be a mapping strictly increasing and continuous with respect to the first variable, and \(\mathcal{A}\)-measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $$F(x)=\int_\Omega F(\tau(x,\omega))\, P(d\omega).$$ We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.
19. Stability aspects of arithmetic functions II
    Acta Arithmetica 139 (2009), 131-146 ResearchGate
Abstract
We deal with the stability problem for arithmetic additive and multiplicative functions in the sense of Hyers and Ulam. In the additive case, the main stability result has the following form: Let an arithmetic function \(f\colon\mathbb{N}\to\mathbb{R}\) satisfy the condition $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\vert f(xy)-f(x)-f(y)\vert\leq\varepsilon.$$ Then there exists a real additive function \(\widetilde{f}\) such that \(\vert f(x)-\widetilde{f}(x)\vert\leq K\varepsilon\) for all \(x\in\mathbb{N}\) with the Kalton-Roberts constant \(K\leq 89/2\).
In the multiplicative case, the main stability assertion is similar: Let an arithmetic function \(f\colon\mathbb{N}\to\mathbb{C}\setminus\{0\}\) satisfy the condition $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\Biggl|\frac{f(xy)}{f(x)f(y)}-1\Biggr|\leq\varepsilon$$ with some \(0\leq\varepsilon<1\). Then there exists an arithmetic multiplicative function \(\widetilde{f}\colon\mathbb{N}\to\mathbb{C}\setminus\{0\}\) such that $$\Biggl|\frac{f(x)}{\widetilde{f}(x)}-1\Biggr|\leq\delta(\varepsilon)\quad\mbox{and}\quad\Biggl|\frac{\widetilde{f}(x)}{f(x)}-1\Biggr|\leq\delta(\varepsilon)\quad\mbox{for }x\in\mathbb{N},$$ where \(\delta(\varepsilon)\) is a non-negative number depending only on \(\varepsilon\). Moreover, \(\delta(\varepsilon)\to 0\) as \(\varepsilon\to 0\).
Unfortunately, as it was pointed out by Professor Andrzej Schinzel, there is a gap in the proof of the first mentioned result, which also impacts the second. In the subsequent work we show that these statements hold true under some additional requirements. Therefore, the two theorems mentioned above have a hypothetical nature.
20. Probability distribution solutions of a general linear equation of infinite order
    Annales Polonici Mathematici 95 (2009), 103-114
    (jointly with J. Morawiec)
Abstract
Let \((\Omega,\mathcal{A},P)\) be a probability space and let \(\tau\colon\mathbb{R}\times\Omega\to\mathbb{R}\) be strictly increasing and continuous with respect to the first variable, and \(\mathcal{A}\)-measurable with respect to the second variable. We obtain a partial characterisation and a uniqueness-type result for solutions of the general linear equation $$F(x)=\int_\Omega F(\tau(x,\omega))\, P(d\omega)$$ in the class of probability distribution functions.
21. An inconsistency equation involving means
    Colloquium Mathematicum 115 (2009), 87-99 ResearchGate
    (jointly with R. Ger)
Abstract
We show that any quasi-arithmetic mean \(A_\varphi\) and any non-quasi-arithmetic mean \(M\) (reasonably regular) are inconsistent in the sense that the only solutions \(f\) of both equations $$f(M(x,y))=A_\varphi(f(x),f(y))$$ and $$f(A_\varphi(x,y))=M(f(x),f(y))$$ are the constant ones.
22. Measurable orthogonally additive functions modulo a discrete subgroup
    Acta Mathematica Hungarica 123 (2009), 239-248
    (jointly with W. Wyrobek)
Abstract
Under appropriate conditions on Abelian topological groups \(G\) and \(H\), an orthogonality \(\bot\subset G^2\) and a \(\sigma\)-algebra \(\mathfrak{M}\) of subsets of \(G\) we decompose an \(\mathfrak{M}\)-measurable function \(f\colon G\to H\) which is orthogonally additive modulo a discrete subgroup \(K\) of \(H\) into its continuous additive and continuous quadratic part (modulo \(K\)).
23. Stability aspects of arithmetic functions
    Acta Arithmetica 132 (2008), 87-98 ResearchGate
Abstract
An arithmetic function \(f\colon\mathbb{N}\to\mathbb{C}\) is called almost additive if for some fixed \(\varepsilon\geq 0\) we have: $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\vert f(xy)-f(x)-f(y)\vert\leq\varepsilon.$$ An arithmetic function \(f\colon\mathbb{N}\to\mathbb{C}\setminus\{0\}\) is called almost multiplicative if for some \(\varepsilon\in [0,1)\) we have: $$x,y\in\mathbb{N},\, (x,y)=1\,\,\Rightarrow\,\,\Biggl|\frac{f(xy)}{f(x)f(y)}-1\Biggr|\leq\varepsilon.$$ Using the Banach limit technique and Ramsey's theorem we derive some stability properties for almost additive and almost multiplicative functions. For instance, we show that for any almost additive function \(f\colon\mathbb{N}\to\mathbb{R}\) satisfying $$\liminf_{x\to\infty}(f(x+1)-f(x))\geq 0$$ there exists a constant \(c\in\mathbb{R}\) such that $$\vert f(x)-c\log x\vert\leq\varepsilon,\quad x\in\mathbb{N}.$$ This assertion generalizes some earlier results due to P. Erdös, I. Kátai and A. Máté.
24. Stability problem for number-theoretically multiplicative functions
    Proceedings of the American Mathematical Society 135 (2007), 2591-2597 ResearchGate
    (jointly with M. Lewicki)
Abstract
We deal with the stability question for multiplicative mappings in the sense of number theory. It turns out that the conditional stability assumption: $$\vert f(xy)-f(x)f(y)\vert\leq\varepsilon\quad\mbox{for relatively prime }x,y$$ implies that \(f\) lies near to some number-theoretically multiplicative function. The domain of \(f\) can be general enough to admit, in special cases, the reduction of our result to the well known J.A. Baker-J. Lawrence-F. Zorzitto superstability theorem.