Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1208.4762

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

H. G. Dales·Tomasz Kania·Tomasz Kochanek·Piotr Koszmider·Niels Jakob Laustsen
View original source ↗Read source document ↗Export metadata ↓

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 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.

Explore related subjects

math.FAmath.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

H. G. Dales, Tomasz Kania, Tomasz Kochanek, Piotr Koszmider, Niels Jakob Laustsen. 2013-09-11. Maximal left ideals of the Banach algebra of bounded operators on a Banach space. https://doi.org/10.4064/sm218-3-3

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Complex Geometric Approach to the Discrete Gabor Transform and Localization Operators on the Flat Torus

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann-Fock spaces can be expressed as theta functions (or equivalently line bundles on Abelian varieties). We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get necessary conditions which hold in higher dimensions and can expand the known results in the one dimensional case, the primary tool being the theorem of the square. We also give an application to asymptotics of restriction operators which arises via the asymptotic behavior of Bergman kernels and Toeplitz operators for high tensor powers of line bundles and find that time frequency restriction operators on the flat torus will exhibit "plunge" behaviors similar to those of time frequency restriction operators in other contexts.

math.FA↗
arXiv2026

On a minimal Andô dilation for a pair of strict contractions

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a commuting pair of strict contractions on a Hilbert space $\mathcal H$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb X$ provided that the function $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.

math.FA↗
arXiv2026

Some properties of Fourier quasicrystals and measures on a strip

We extend certain results of the theory of Fourier quasicrystals on the real line to the case of a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures $μ$ on a strip whose Fourier transform is a pure point measure $\hatμ=\sum_{γ\inΓ}b_γδ_γ$ (as usual, $δ_γ$ is the unit mass at the point $γ$). We prove that the measure $ν=\sum_{γ\inΓ}|b_γ|^2δ_γ$ has the exponential growth. Moreover, if for some $η>0$ the points of $Γ$ in every interval of length $η$ are linearly independent over integers, then the measure $\hatμ$ also has the exponential growth.

math.FA↗