arXiv subjects

Bence Horváth

Publications and source records attributed to Bence Horváth.

11 recordsLinked to original sources

arXiv2026

Twisting exponential spectra

Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of $ab$ and $ba$ differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain--Delbaen space $X$ obtained from Motakis' construction with Calkin algebra isomorphic as a Banach algebra to $C(S^4)$, we construct $S,T\in\B(X\oplus X)$ such that \[ \frac12\in\varepsilon_{\B(X\oplus X)}(ST) \quad\text{and}\quad \frac12\notin\varepsilon_{\B(X\oplus X)}(TS). \] Moreover, $\ltsr\B(X\oplus X)=\rtsr\B(X\oplus X)=2$, which is the least possible stable rank for such an example. More generally, if $K$ is compact metrisable and $X$ is any space arising from Motakis' construction for $C(K)$, then, for every $n\geqslant1$, \[ \ltsr\B(X^n)=\rtsr\B(X^n)= \begin{cases} \left\lceil \lfloor\dim K/2\rfloor/n\right\rceil+1,&\dim K<\infty, \infty,&\dim K=\infty. \end{cases} \] This also gives counterexamples of every finite stable rank at least two and of infinite stable rank. Finally, we relate every failure of exponential-spectral commutativity to the kernel of the first matrix stabilisation map on the index group.

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arXiv2026

Pure infiniteness and primary factorisation

We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}_E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$. For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar. We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E_{\operatorname{R}}$ does not have the uniform PFP.

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arXiv2024

Kernels of operators on Banach spaces induced by almost disjoint families

Let~$\mathcal{A}$ be an almost disjoint family of subsets of an infinite set~$Γ$, and denote by~$X_{\mathcal{A}}$ the closed subspace of~$\ell_\infty(Γ)$ spanned by the indicator functions of intersections of finitely many sets in~$\mathcal{A}$. We show that if~$\mathcal{A}$ has cardinality greater than~$Γ$, then the closed subspace of~$X_{\mathcal{A}}$ spanned by the indicator functions of sets of the form $\bigcap_{j=1}^{n+1}A_j$, where $n\in\N$ and $A_1,\ldots,A_{n+1}\in\mathcal{A}$ are distinct, cannot be the kernel of any bounded operator \mbox{$X_{\mathcal{A}}\rightarrow \ell_{\infty}(Γ)$}. As a consequence, we deduce that the subspace \[ \bigl\{ x\in \ell_{\infty}(Γ) : \text{the set}\ \{γ\in Γ: \lvert x(γ)\rvert > \varepsilon \}\ \text{has cardinality smaller than}\ Γ \text{for every}\ \varepsilon>0\bigr\} \] of~$\ell_\infty(Γ)$ is not the kernel of any bounded operator on~$\ell_\infty(Γ)$; this generalises results of Kalton and of Pełczyński and Sudakov. The situation is more complex for the Banach space~$\ell_\infty^c(Γ)$ of countably supported, bounded functions defined on an uncountable set~$Γ$. We show that it is undecidable in \textsf{ZFC} whether every bounded operator on~$\ell_\infty^c(ω_1)$ which vanishes on~$c_0(ω_1)$ must vanish on a subspace of the form~$\ell_\infty^c(A)$ for some uncountable subset~$A$ of~$ω_1$.

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arXiv2022

A purely infinite Cuntz-like Banach $*$-algebra with no purely infinite ultrapowers

We continue our investigation, from \cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a "Cuntz-like" Banach $*$-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algebra is a naturally occurring analogue of the Cuntz algebra, and of the $L^p$-analogues introduced by Phillips. However, our proof of being purely infinite is combinatorial, but direct, and so differs from existing proofs. We show that there are non-zero traces on our algebra, which in particular implies that our algebra is not isomorphic to any of the $L^p$-analogues of the Cuntz algebra.

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arXiv2022

Approximately multiplicative maps between algebras of bounded operators on Banach spaces

We show that for any separable reflexive Banach space $X$ and a large class of Banach spaces $E$, including those with a subsymmetric shrinking basis but also all spaces $L_p$ for $1\leq p \leq \infty$, every bounded linear map ${\mathcal B}(E)\to {\mathcal B}(X)$ which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism ${\mathcal B}(E)\to {\mathcal B}(X)$. That is, the pair $({\mathcal B}(E), {\mathcal B}(X))$ has the AMNM property in the sense of Johnson (\textit{J.~London Math.\ Soc.} 1988). Previously this was only known for $E=X=\ell_p$ with $1<p<\infty$; even for those cases, we improve on the previous methods and obtain better constants in various estimates. A crucial role in our approach is played by a new result, motivated by cohomological techniques, which establishes AMNM properties relative to an amenable subalgebra; this generalizes a theorem of Johnson (\textit{op cit.}).

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arXiv2021

Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

A remarkable result of Molnár [Proc. Amer. Math. Soc., 126 (1998), 853-861] states that automorphisms of the algebra of operators acting on a separable Hilbert space is stable under "small" perturbations. More precisely, if $ϕ,ψ$ are endomorphisms of $\mathcal{B}(\mathcal{H})$ such that $\|ϕ(A)-ψ(A)\|<\|A\|$ and $ψ$ is surjective then so is $ϕ$. The aim of this paper is to extend this result to a larger class of Banach spaces including $\ell_p$ and $L_p$ spaces ($1<p<+\infty$). En route to the proof we show that for any Banach space $X$ from the above class all faithful, unital, separable, reflexive representations of $\mathcal B (X)$ which preserve rank one operators are in fact isomorphisms.

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arXiv2021

Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces

Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may be found of the form $C(X)$ for a compact space $X$, $\ell_1(G)$ for some torsion-free Abelian group, and a~simple, unital AF $C^*$-algebra. Extensions to higher densities are also presented.

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arXiv2020

Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective

We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on $X$, to $\mathscr{B}(Y)$, where $X$ is one of the following \emph{long} sequence spaces: $c_0(λ)$, $\ell_{\infty}^c(λ)$, and $\ell_p(λ)$ ($1 \leqslant p < \infty$) and $Y$ is arbitrary. \textit{En route} to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the `sequential strong operator topology'.

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arXiv2020

Ring-theoretic (in)finiteness in reduced products of Banach algebras

We study ring-theoretic (in)finiteness properties -- such as \emph{Dedekind-finiteness} and \emph{proper infiniteness} -- of ultraproducts (and more generally, reduced products) of Banach algebras. Whilst we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the $C^*$-algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if "ultrafilter many" of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of Łoś's Theorem; the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of \emph{fixed} bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for $C^*$-algebras. Finally the related notion of having \textit{stable rank one} is also studied for ultraproducts.

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arXiv2019

When are full representations of algebras of operators on Banach spaces automatically faithful?

We examine the phenomenon when surjective algebra homomorphisms between algebras of operators on Banach spaces are automatically injective. In the first part of the paper we shall show that for certain Banach spaces $X$ the following property holds: For every non-zero Banach space $Y$ every surjective algebra homomorphism $ψ: \, \mathcal{B}(X) \rightarrow \mathcal{B}(Y)$ is automatically injective. In the second part of the paper we consider the question in the opposite direction: Building on the work of Kania, Koszmider and Laustsen \textit{(Trans. London Math. Soc., 2014)} we show that for every separable, reflexive Banach space $X$ there is a Banach space $Y_X$ and a surjective but not injective algebra homomorphism $ψ: \, \mathcal{B}(Y_X) \rightarrow \mathcal{B}(X)$.

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arXiv2019

A Banach space whose algebra of operators is Dedekind-finite but it does not have stable rank one

In this note we examine the connection between the stable rank one and Dedekind-finite property of the algebra of operators on a Banach space $X$. We show that for the indecomposable but not hereditarily indecomposable Banach space $X_{\infty}$ constructed by Tarbard (Ph.D. Thesis, University of Oxford, 2013), the algebra of operators $B(X_{\infty})$ is Dedekind-finite but does not have stable rank one. While this sheds some light on the Banach space structure of $X_{\infty}$ itself, we observe that the indecomposable but not hereditarily indecomposable Banach space constructed by Gowers and Maurey (Math. Ann., 1997) does not possess this property.

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