Uniform spectral radius and compact Gelfand transform

Alexandru Aleman, and Anders Dahlner. "Uniform spectral radius and compact Gelfand transform." Studia Mathematica 172.1 (2006): 25-46. <http://eudml.org/doc/284901>.

@article{AlexandruAleman2006,
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 ↦ x̂ are: (i) Is $K_\{ν\} = sup\{||(e-x)^\{-1\}||_\{p\}: x ∈ A, ||x||_\{p\} ≤ 1, max|x̂| ≤ ν\}$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_\{δ\} = sup\{||x^\{-1\}||_\{p\}: x ∈ A, ||x||_\{p\} ≤ 1, min|x̂| ≥ δ\}$ 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 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 δ ∈ (0,1) if and only if $r_\{∞\}(A) = 0$. Finally, we specialize to semisimple Beurling type algebras $ℓ^\{p\}_\{ω\}()$, where 0 < p < 1 and = ℕ or = ℤ. We show that the number $r_\{∞\}(ℓ^\{p\}_\{ω\}())$ 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 = {Alexandru Aleman, Anders Dahlner},
journal = {Studia Mathematica},
keywords = {uniform spectral radius; norm controlled inversion; bounded inverse property; invisible spectrum; quasi-Banach algebra},
language = {eng},
number = {1},
pages = {25-46},
title = {Uniform spectral radius and compact Gelfand transform},
url = {http://eudml.org/doc/284901},
volume = {172},
year = {2006},
}

TY - JOUR
AU - Alexandru Aleman
AU - Anders Dahlner
TI - Uniform spectral radius and compact Gelfand transform
JO - Studia Mathematica
PY - 2006
VL - 172
IS - 1
SP - 25
EP - 46
AB - 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 $K_{ν} = sup{||(e-x)^{-1}||_{p}: x ∈ A, ||x||_{p} ≤ 1, max|x̂| ≤ ν}$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{δ} = sup{||x^{-1}||_{p}: x ∈ A, ||x||_{p} ≤ 1, min|x̂| ≥ δ}$ 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 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 δ ∈ (0,1) if and only if $r_{∞}(A) = 0$. Finally, we specialize to semisimple Beurling type algebras $ℓ^{p}_{ω}()$, where 0 < p < 1 and = ℕ or = ℤ. We show that the number $r_{∞}(ℓ^{p}_{ω}())$ 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].
LA - eng
KW - uniform spectral radius; norm controlled inversion; bounded inverse property; invisible spectrum; quasi-Banach algebra
UR - http://eudml.org/doc/284901
ER -