Approximately multiplicative maps between algebras of bounded operators on Banach spaces

TL;DR

This paper proves that approximately multiplicative maps between certain operator algebras on Banach spaces are close to true homomorphisms, extending known results and improving constants for a broad class of spaces.

Contribution

It establishes the AMNM property for pairs of operator algebras on Banach spaces, generalizing previous results and introducing new cohomological techniques.

Findings

Approximately multiplicative maps are close to homomorphisms in operator norm.

The AMNM property holds for a wide class of Banach spaces including $L_p$ spaces.

New cohomological methods are developed to analyze AMNM properties relative to amenable subalgebras.

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.}).