Annals of Functional Analysis

Polynomial identification in uniform and operator algebras

Dillon Ethier, Tova Lindberg, and Aaron Luttman

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Let $\A$ be a unital Banach algebra, and denote the spectral radius of $f\in\A$ by $\rho(f)$. If $\A$ is a uniform algebra and $\rho(fh+1)=\rho(gh+1)$ for all $h\in\A$, then it can be shown that $f=g$, a result that also carries in algebras of bounded linear operators on Banach spaces. On the other hand $\rho(fh)=\rho(gh)$ does not imply $f=g$ in any unital algebra, marking a distinction between the polynomials $p(z,w)=zw+1$ and $p(z,w)=zw$. Such results are known as spectral identification lemmas, and in this work we demonstrate first- and second-degree polynomials of two variables that lead to identification via the spectral radius, peripheral spectrum, or full spectrum in uniform algebras and in algebras of bounded linear operators on Banach spaces. The primary usefulness of identification lemmas is to determine the injectivity of a class of mappings that preserve portions of the spectrum, and results corresponding to the given identifications are also presented.

Article information

Ann. Funct. Anal., Volume 1, Number 1 (2010), 105-122 .

First available in Project Euclid: 12 May 2014

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Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 47L10: Algebras of operators on Banach spaces and other topological linear spaces 47A65: Structure theory 46J20: Ideals, maximal ideals, boundaries 46H20: Structure, classification of topological algebras 47C05: Operators in algebras

Uniform algebras standard operator algebras polynomial identification peripheral spectrum spectral preserver problems


Ethier, Dillon; Lindberg, Tova; Luttman, Aaron. Polynomial identification in uniform and operator algebras. Ann. Funct. Anal. 1 (2010), no. 1, 105--122. doi:10.15352/afa/1399900997.

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