Almost multiplicative functionals

Krzysztof Jarosz

Studia Mathematica (1997)

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

Abstract

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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 ∞ / B H ∞ is f-stable if and only if B is a product of finitely many interpolating Blaschke products.

How to cite

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Jarosz, 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 -

References

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