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Uniform spectral radius and compact Gelfand transform

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Author

Summary, in English

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 bar right arrow (x) over cap are: (i) Is K-nu = sup{parallel to(e - x)(-1)parallel to(p) : x is an element of A, parallel to x parallel to(p) <= 1, max (x) over cap <= nu} bounded, where nu is an element of (0, 1)? (ii) For which delta is an element of (0, 1) is C-delta = sup{parallel to x(-1)parallel to(p) : x is an element of A, parallel to x parallel to(p) <= 1, min (x) over cap >= delta} bounded? Both questions are related to a "uniform spectral radius" of the algebra, r(infinity)(A), introduced by Bjork. Question (i) has an affirmative answer if and only if r(infinity)(A) < 1, and this result is extended to more general nonlinear extremal problems of this type. Question (ii) is more difficult, but it can also be related to the uniform spectral radius. For algebras with compact Gelfand transform we prove that the answer is "yes" for all delta is an element of (0, 1) if and only if r(infinity)(A) = 0. Finally, we specialize to semisimple Beurling type algebras l(w)(p)(D), where 0 < p < 1 and D = N or D = Z. We show that the number r(infinity)(l(w)(p)(D)) can be effectively computed in terms of the underlying weight. In particular, this solves questions (i) and (ii) for many of these algebras. We also construct weights such that the corresponding Beurling algebra has a compact Gelfand transform, but the uniform spectral radius equals an arbitrary given number in (0, 1].

Publishing year

2006

Language

English

Pages

25-46

Publication/Series

Studia Mathematica

Volume

172

Issue

1

Document type

Journal article

Publisher

Polish Academy of Sciences

Topic

  • Mathematics

Keywords

  • quasi-Banach algebras
  • property
  • bounded inverse
  • uniform spectral radius
  • norm controlled inversion
  • invisible spectrum

Status

Published

ISBN/ISSN/Other

  • ISSN: 0039-3223