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Polynomial identification in uniform and operator algebras. (English) Zbl 1220.46032

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 𝒜 be a uniform algebra on a compact Hausdorff space K, and let f,g be two elements of 𝒜.

Among other results, the authors show that, if there exist α,β{0} and γ such that αf+βh+γ=αg+βh+γ for all -peaking functions h𝒜, then f=g. Recall that a nonzero element h𝒜 is called a -peaking function if its peripheral spectrum is a singleton.

In the second part, the authors consider standard operator algebras. Let 𝒜 be a unital standard operator algebra on a Banach space X and let A,B be in 𝒜. The authors show that, if there exist α,β{0} and γ such that ρ(αA+βT+γ)=ρ(αB+βT+γ) for all T𝒜, where ρ(·) denotes the spectral radius, then A=B. Other identification criteria are given.

MSC:
46J10Banach algebras of continuous functions, function algebras
47L10Algebras of operators on Banach spaces and other topological linear spaces
47A65Structure theory of linear operators
46J20Ideals, maximal ideals, boundaries of commutative topological algebras
46H20Structure and classification of topological algebras
47C05Operators in topological algebras