Uniform spectral radius and compact Gelfand transform
Aleman, Alexandru LU and Dahlner, Anders LU
(2006)
In Studia Mathematica
172(1).
p.25-46
- Abstract
- 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... (More)
- 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]. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/410611
- author
- Aleman, Alexandru
LU
and Dahlner, Anders
LU
- organization
- publishing date
- 2006
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- quasi-Banach algebras, property, bounded inverse, uniform spectral radius, norm controlled inversion, invisible spectrum
- in
- Studia Mathematica
- volume
- 172
- issue
- 1
- pages
- 25 - 46
- publisher
- Polish Academy of Sciences
- external identifiers
-
- wos:000237152900002
- scopus:33744980657
- ISSN
- 0039-3223
- language
- English
- LU publication?
- yes
- id
- 6fa12639-727d-4935-b273-cfa5c58e20e5 (old id 410611)
- alternative location
- http://journals.impan.gov.pl/cgi-bin/sm/pdf?sm172-1-02
- date added to LUP
- 2016-04-01 15:26:34
- date last changed
- 2023-04-20 15:26:12
@article{6fa12639-727d-4935-b273-cfa5c58e20e5,
abstract = {{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].}},
author = {{Aleman, Alexandru and Dahlner, Anders}},
issn = {{0039-3223}},
keywords = {{quasi-Banach algebras; property; bounded inverse; uniform spectral radius; norm controlled inversion; invisible spectrum}},
language = {{eng}},
number = {{1}},
pages = {{25--46}},
publisher = {{Polish Academy of Sciences}},
series = {{Studia Mathematica}},
title = {{Uniform spectral radius and compact Gelfand transform}},
url = {{http://journals.impan.gov.pl/cgi-bin/sm/pdf?sm172-1-02}},
volume = {{172}},
year = {{2006}},
}