Uniform spectral radius and compact Gelfand transform
(2006) In Studia Mathematica 172(1). p.2546 Abstract
 We consider the quantization of inversion in commutative pnormed quasiBanach 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 Knu = 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 Cdelta = 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 pnormed quasiBanach 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 Knu = 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 Cdelta = 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
 quasiBanach 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
 00393223
 language
 English
 LU publication?
 yes
 id
 6fa12639727d4935b273cfa5c58e20e5 (old id 410611)
 alternative location
 http://journals.impan.gov.pl/cgibin/sm/pdf?sm172102
 date added to LUP
 20160401 15:26:34
 date last changed
 20210630 04:00:15
@article{6fa12639727d4935b273cfa5c58e20e5, abstract = {We consider the quantization of inversion in commutative pnormed quasiBanach 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 Knu = 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 Cdelta = 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 = {00393223}, language = {eng}, number = {1}, pages = {2546}, publisher = {Polish Academy of Sciences}, series = {Studia Mathematica}, title = {Uniform spectral radius and compact Gelfand transform}, url = {http://journals.impan.gov.pl/cgibin/sm/pdf?sm172102}, volume = {172}, year = {2006}, }