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Zbl 1220.46032
Ethier, Dillon; Lindberg, Tova; Luttman, Aaron
Polynomial identification in uniform and operator algebras.
(English)
[J] Ann. Funct. Anal. AFA 1, No. 1, 105-122, electronic only (2010). ISSN 2008-8752/e

The authors establish various criteria for the equality of two elements in some unital Banach algebras. The first part of the paper is devoted to uniform algebras. Let $\cal A$ be a uniform algebra on a compact Hausdorff space $K$, and let $f,g$ be two elements of $\cal A$. Among other results, the authors show that, if there exist $\alpha, \beta \in \mathbb C \setminus \{ 0 \}$ and $\gamma \in \mathbb C$ such that $\| \alpha f + \beta h + \gamma \| = \| \alpha g + \beta h + \gamma \|$ for all $\mathbb C$-peaking functions $h \in \cal A$, then $f=g$. Recall that a nonzero element $h \in \cal A$ is called a $\mathbb C$-peaking function if its peripheral spectrum is a singleton. \par In the second part, the authors consider standard operator algebras. Let $\cal A$ be a unital standard operator algebra on a Banach space $X$ and let $A, B$ be in $\cal A$. The authors show that, if there exist $\alpha, \beta \in \mathbb C \setminus \{ 0 \}$ and $\gamma \in \mathbb C$ such that $\rho (\alpha A+ \beta T+ \gamma)= \rho (\alpha B+ \beta T+ \gamma)$ for all $T \in \cal A$, where $\rho (\cdot)$ denotes the spectral radius, then $A=B$. Other identification criteria are given.
MSC 2000:
*46J10 Banach algebras of continuous functions
47L10 Algebras of operators on Banach spaces, etc.
47A65 Structure theory of linear operators
46J20 Ideals of commutative topological algebras
46H20 Structure and classification of topological algebras
47C05 Operators in topological algebras

Keywords: uniform algebras; standard operator algebras; peripheral spectrum; polynomial identification; spectral preserver problems

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