Uniform spectral radius and compact Gelfand transform

Alexandru Aleman¹ ; Anders Dahlner¹

¹ Centre for Mathematical Sciences Mathematics (Faculty of Science) University of Lund, Box 118 SE-221 00 Lund, Sweden

Studia Mathematica, Tome 172 (2006) no. 1, pp. 25-46

Résumé

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 $x\mapsto\widehat{x}$ are: (i) Is $K_\nu=\sup\{\|(e-x)^{-1}\|_p:x\in A, \|x\|_p\leq 1,\, \max|\widehat{x}|\leq\nu\}$ bounded, where $\nu\in(0,1)$? (ii) For which $\delta\in(0,1)$ is $C_\delta=\sup\{\|x^{-1}\|_p:x\in A,\, \|x\|_p\leq1,\, \min|\widehat{x}|\geq\delta\}$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_\infty(A)$, introduced by Björk. Question (i) has an affirmative answer if and only if $r_\infty(A) 1$, 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 $\delta\in(0,1)$ if and only if $r_\infty(A)=0$. Finally, we specialize to semisimple Beurling type algebras $\ell^p_\omega ({\cal D})$, where $0 p 1$ and ${\cal D}={\mathbb N}$ or ${\cal D}={\mathbb Z}$. We show that the number $r_\infty(\ell^p_\omega ({\cal D}))$ 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 $(0,1]$.

Détail

DOI : 10.4064/sm172-1-2

Keywords: consider quantization inversion commutative p normed quasi banach algebras unit standard questions considered algebra unit gelfand transform mapsto widehat sup e x leq max widehat leq bounded where which delta delta sup leq min widehat geq delta bounded questions related uniform spectral radius algebra infty introduced question has affirmative answer only infty result extended general nonlinear extremal problems type question difficult related uniform spectral radius algebras compact gelfand transform prove answer yes delta only infty finally specialize semisimple beurling type algebras ell omega cal where cal mathbb cal mathbb number infty ell omega cal effectively computed terms underlying weight particular solves questions many these algebras construct weights corresponding beurling algebra has compact gelfand transform uniform spectral radius equals arbitrary given number