Uniform spectral radius and compact Gelfand transform
Volume 172 / 2006
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 $x\mapsto\widehat{x}$ 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 $(0,1]$.
