Approximately multiplicative maps between algebras of bounded operators on Banach spaces

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 $1\le p\le \infty$, 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.}).

v1: AMS-LaTeX, 30 pages. Submitted for publication. v2: incorporates revisions based on feedback from referee; minor updates/corrections to bibliography. Final accepted version, to appear in Trans. Amer. Math. Soc

References in corpus (3)

• Conditions implying the uniqueness of the weak${}^{\ast }$-topology on certain group algebras
• Approximately multiplicative maps from weighted semilattice algebras
• Perturbations of surjective homomorphisms between algebras of operators on Banach spaces