Rights statement: First published in American Mathematical Society in volume 375, number 10, October 2022, published by the American Mathematical Society. © 2021 American Mathematical Society.
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Final published version
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| <mark>Journal publication date</mark> | 31/10/2022 |
|---|---|
| <mark>Journal</mark> | Transactions of the American Mathematical Society |
| Issue number | 10 |
| Volume | 375 |
| Number of pages | 27 |
| Pages (from-to) | 7121-7147 |
| Publication Status | Published |
| Early online date | 29/07/22 |
| <mark>Original language</mark> | English |
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.).