Approximately multiplicative maps between algebras of bounded operators on Banach spaces

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.).