Approximately multiplicative maps between algebras of bounded operators on Banach spaces

Abstract

We show that for any separable reflexive Banach space X X and a large class of Banach spaces E E , including those with a subsymmetric shrinking basis but also all spaces L p [ 0 , 1 ] L_p[0,1] for 1 ≤ p ≤ ∞ 1\le p \le \infty , every bounded linear map B ( E ) → B ( X ) \mathcal {B}(E)\to \mathcal {B}(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism B ( E ) → B ( X ) \mathcal {B}(E)\to \mathcal {B}(X) . That is, the pair ( B ( E ) , B ( X ) ) (\mathcal {B}(E),\mathcal {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 E=X=\ell _p with 1 > p > ∞ 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 (op cit.).