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Uniform spectral radius and compact Gelfand transform

Alexandru AlemanAnders Dahlner — 2006

Studia Mathematica

We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is K ν = s u p | | ( e - x ) - 1 | | p : x A , | | x | | p 1 , m a x | x ̂ | ν bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is C δ = s u p | | x - 1 | | p : x A , | | x | | p 1 , m i n | x ̂ | δ bounded? Both questions are related to a “uniform spectral radius” of the algebra, r ( A ) , introduced by Björk. Question (i) has an affirmative answer if and only if r ( A ) < 1 , and this result is extended to more general nonlinear extremal problems...

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