Hermitian powers: A Müntz theorem and extremal algebras

M. J. Crabb¹ ; J. Duncan² ; C. M. McGregor¹ ; T. J. Ransford³

¹ Department of Mathematics University of Glasgow Glasgow G12 8QW, U.K.
² Department of Mathematical Sciences University of Arkansas Fayetteville, AR 72701-1201, U.S.A.
³ Department of Mathematics and Statistics Laval University Québec, Canada G1K 7P4

Studia Mathematica, Tome 146 (2001) no. 1, pp. 83-97

Résumé

Given ${\mathbb S}\subset {\mathbb N}$, let $\widehat {{\mathbb S}}$ be the set of all positive integers $m$ for which $h^m$ is hermitian whenever $h$ is an element of a complex unital Banach algebra $A$ with $h^n$ hermitian for each $n\in {\mathbb S}$. We attempt to characterize when (i) $\widehat {{\mathbb S}}={\mathbb N}$, or (ii) $\widehat {{\mathbb S}}={\mathbb S}$. A key tool is a Müntz-type theorem which gives remarkable conclusions when $1\in {\mathbb S}$ and $\sum \{ 1/n:n\in {\mathbb S}\} $ diverges. The set $\widehat {{\mathbb S}}$ is determined by a single extremal Banach algebra $\mathop {\rm Ea}\nolimits ({\mathbb S})$. We describe this extremal algebra for various ${\mathbb S}$.

Détail

DOI : 10.4064/sm146-1-6

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