Hermitian powers: A Müntz theorem and extremal algebras

Volume 146 / 2001

M. J. Crabb, J. Duncan, C. M. McGregor, T. J. Ransford Studia Mathematica 146 (2001), 83-97 MSC: Primary 46H05. DOI: 10.4064/sm146-1-6

Abstract

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 ${\mathbb S}$.

Authors

  • M. J. CrabbDepartment of Mathematics
    University of Glasgow
    Glasgow G12 8QW, U.K.
    e-mail
  • J. DuncanDepartment of Mathematical Sciences
    University of Arkansas
    Fayetteville, AR 72701-1201, U.S.A.
    e-mail
  • C. M. McGregorDepartment of Mathematics
    University of Glasgow
    Glasgow G12 8QW, U.K.
    e-mail
  • T. J. RansfordDepartment of Mathematics and Statistics
    Laval University
    Québec, Canada G1K 7P4
    e-mail

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