Home > Research > Publications & Outputs > Approximately multiplicative maps between algeb...

Electronic data

  • AMNM-BE_arX2

    Accepted author manuscript, 540 KB, PDF document

    Available under license: CC BY-NC-ND: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

View graph of relations

Approximately multiplicative maps between algebras of bounded operators on Banach spaces

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Forthcoming
<mark>Journal publication date</mark>23/02/2022
<mark>Journal</mark>Transactions of the American Mathematical Society
Publication StatusAccepted/In press
<mark>Original language</mark>English

Abstract

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.}).