- AMNM-BE_arX2
**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.Accepted author manuscript, 540 KB, PDF document

Available under license: CC BY-NC-ND: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

- https://www.ams.org/journals/tran/0000-000-00/S0002-9947-2022-08687-6/
Final published version

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

In: Transactions of the American Mathematical Society, Vol. 375, No. 10, 31.10.2022, p. 7121-7147.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Choi, Y, Horvath, B & Laustsen, N 2022, 'Approximately multiplicative maps between algebras of bounded operators on Banach spaces', *Transactions of the American Mathematical Society*, vol. 375, no. 10, pp. 7121-7147. https://doi.org/10.1090/tran/8687

Choi, Y., Horvath, B., & Laustsen, N. (2022). Approximately multiplicative maps between algebras of bounded operators on Banach spaces. *Transactions of the American Mathematical Society*, *375*(10), 7121-7147. https://doi.org/10.1090/tran/8687

Choi Y, Horvath B, Laustsen N. Approximately multiplicative maps between algebras of bounded operators on Banach spaces. Transactions of the American Mathematical Society. 2022 Oct 31;375(10):7121-7147. Epub 2022 Jul 29. doi: 10.1090/tran/8687

@article{76537a4649d54764b739116d99229ec0,

title = "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[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.). ",

keywords = "AMNM, Algebra of bounded operators, Banach space, Hochschild cohomology, Ulam stability, amenable Banach algebra, approximately multiplicative, homomorphism, perturbation",

author = "Yemon Choi and Bence Horvath and Niels Laustsen",

year = "2022",

month = oct,

day = "31",

doi = "10.1090/tran/8687",

language = "English",

volume = "375",

pages = "7121--7147",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "10",

}

TY - JOUR

T1 - Approximately multiplicative maps between algebras of bounded operators on Banach spaces

AU - Choi, Yemon

AU - Horvath, Bence

AU - Laustsen, Niels

PY - 2022/10/31

Y1 - 2022/10/31

N2 - 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.).

AB - 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.).

KW - AMNM

KW - Algebra of bounded operators

KW - Banach space

KW - Hochschild cohomology

KW - Ulam stability

KW - amenable Banach algebra

KW - approximately multiplicative

KW - homomorphism

KW - perturbation

U2 - 10.1090/tran/8687

DO - 10.1090/tran/8687

M3 - Journal article

VL - 375

SP - 7121

EP - 7147

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -