Gelfand-Hille type theorems in ordered Banach algebras

Gelfand-Hille type theorems in ordered Banach algebras We consider the Gelfand-Hille Theorems, specifically conditions under which an element in an ordered Banach algebra (A,C) with spectrum {1} is the identity of the algebra. In particular we show that for $$x,x^{-1} \in C$$ , where C is a closed normal algebra cone, if $$\sigma(x) = \{1\}$$ and x is doubly Abel bounded then x = 1. Furthermore in the case where $$\sigma(x) = \{1\}$$ and C is a closed proper algebra cone, then x = 1 if and only if x L is Abel bounded and $$x^N \geq 1$$ for some $$L,N \in \mathbb{N}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Gelfand-Hille type theorems in ordered Banach algebras

, Volume 13 (1) – Oct 28, 2008
12 pages

/lp/springer_journal/gelfand-hille-type-theorems-in-ordered-banach-algebras-HAmw6P82ZY
Publisher
Birkhäuser-Verlag
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2200-4
Publisher site
See Article on Publisher Site

Abstract

We consider the Gelfand-Hille Theorems, specifically conditions under which an element in an ordered Banach algebra (A,C) with spectrum {1} is the identity of the algebra. In particular we show that for $$x,x^{-1} \in C$$ , where C is a closed normal algebra cone, if $$\sigma(x) = \{1\}$$ and x is doubly Abel bounded then x = 1. Furthermore in the case where $$\sigma(x) = \{1\}$$ and C is a closed proper algebra cone, then x = 1 if and only if x L is Abel bounded and $$x^N \geq 1$$ for some $$L,N \in \mathbb{N}$$ .

Journal

PositivitySpringer Journals

Published: Oct 28, 2008

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. Organize your research It’s easy to organize your research with our built-in tools. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from