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### Uniform spectral radius and compact Gelfand transform

and (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)
author
and
organization
publishing date
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
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
2016-04-01 15:26:34
date last changed
2021-06-30 04:00:15
```@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) &lt;= 1, max (x) over cap &lt;= 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) &lt;= 1, min (x) over cap &gt;= 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) &lt; 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 &lt; p &lt; 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},
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},
}

```