Choi, Yemon and Horvath, Bence and Laustsen, Niels
(2022)
*Approximately multiplicative maps between algebras of bounded operators on Banach spaces.*
Transactions of the American Mathematical Society, 375 (10).
pp. 7121-7147.
ISSN 0002-9947

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## 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[0, 1] for 1 ≤ p ≤ ∞, every bounded linear map B(E) → B(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism B(E) → B(X). That is, the pair (B(E), B(X)) has the AMNM property in the sense of Johnson [J. London Math. Soc. (2) 37 (1988), pp. 294–316]. Previously this was only known for E = X = ℓ p with 1 < p < ∞; 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 (op cit.).