Almost multiplicative functionals

Krzysztof Jarosz

ArticleOriginal scientific text

Title

Almost multiplicative functionals

Authors ^1,²

Affiliations

Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43403, U.S.A.
Department of Mathematics, Southern Illinois University at Edwardsville, Edwardsville, Illinois 62026, U.S.A.

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 \subset ℂ^{n}

, are f-stable. We show that, for a Blaschke product B, the quotient algebra

\frac{H^{\infty}}{B} H^{\infty}

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

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Title

Almost multiplicative functionals

Affiliations

Abstract

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