# Almost multiplicative functionals

Studia Mathematica (1997)

- Volume: 124, Issue: 1, page 37-58
- ISSN: 0039-3223

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topJarosz, Krzysztof. "Almost multiplicative functionals." Studia Mathematica 124.1 (1997): 37-58. <http://eudml.org/doc/216396>.

@article{Jarosz1997,

abstract = {A linear functional F on a Banach algebra A is almost multiplicative if |F(ab) - F(a)F(b)| ≤ δ∥a∥ · ∥b∥ for a,b ∈ A, for a small constant δ. An algebra is called functionally stable or f-stable if any almost multiplicative functional is close to a multiplicative one. The question whether an algebra is f-stable can be interpreted as a question whether A lacks an almost corona, that is, a set of almost multiplicative functionals far from the set of multiplicative functionals. In this paper we discuss f-stability for general uniform algebras; we prove that any uniform algebra with one generator as well as some algebras of the form R(K), K ⊂ ℂ, and A(Ω), $Ω ⊂ ℂ^\{n\}$, are f-stable. We show that, for a Blaschke product B, the quotient algebra $H^\{∞\}/BH^\{∞\}$ is f-stable if and only if B is a product of finitely many interpolating Blaschke products.},

author = {Jarosz, Krzysztof},

journal = {Studia Mathematica},

keywords = {Banach algebras; almost multiplicative functionals; functionally stable; -stable; AMNM algebras; uniform algebra; Banach algebras of rational or analytic functions; Blaschke product},

language = {eng},

number = {1},

pages = {37-58},

title = {Almost multiplicative functionals},

url = {http://eudml.org/doc/216396},

volume = {124},

year = {1997},

}

TY - JOUR

AU - Jarosz, Krzysztof

TI - Almost multiplicative functionals

JO - Studia Mathematica

PY - 1997

VL - 124

IS - 1

SP - 37

EP - 58

AB - A linear functional F on a Banach algebra A is almost multiplicative if |F(ab) - F(a)F(b)| ≤ δ∥a∥ · ∥b∥ for a,b ∈ A, for a small constant δ. An algebra is called functionally stable or f-stable if any almost multiplicative functional is close to a multiplicative one. The question whether an algebra is f-stable can be interpreted as a question whether A lacks an almost corona, that is, a set of almost multiplicative functionals far from the set of multiplicative functionals. In this paper we discuss f-stability for general uniform algebras; we prove that any uniform algebra with one generator as well as some algebras of the form R(K), K ⊂ ℂ, and A(Ω), $Ω ⊂ ℂ^{n}$, are f-stable. We show that, for a Blaschke product B, the quotient algebra $H^{∞}/BH^{∞}$ is f-stable if and only if B is a product of finitely many interpolating Blaschke products.

LA - eng

KW - Banach algebras; almost multiplicative functionals; functionally stable; -stable; AMNM algebras; uniform algebra; Banach algebras of rational or analytic functions; Blaschke product

UR - http://eudml.org/doc/216396

ER -

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