# 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
Springer Journals
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

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