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    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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Approximately multiplicative maps between algebras of bounded operators on Banach spaces

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

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Approximately multiplicative maps between algebras of bounded operators on Banach spaces. / Choi, Yemon; Horvath, Bence; Laustsen, Niels.
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

Harvard

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

APA

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

Vancouver

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

Author

Choi, Yemon ; Horvath, Bence ; Laustsen, Niels. / Approximately multiplicative maps between algebras of bounded operators on Banach spaces. In: Transactions of the American Mathematical Society. 2022 ; Vol. 375, No. 10. pp. 7121-7147.

Bibtex

@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",

}

RIS

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 -