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- Title
Gelfand-Hille type theorems in ordered Banach algebras.
- Authors
Braatvedt, Gareth; Brits, Rudi; Raubenheimer, Heinrich
- 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 $$L,N \in \mathbb{N}$$ .
- Subjects
BANACH algebras; GELFAND-Naimark theorem; SPECTRAL geometry; NORMAL numbers; CONES (Operator theory)
- Publication
Positivity, 2009, Vol 13, Issue 1, p39
- ISSN
1385-1292
- Publication type
Article
- DOI
10.1007/s11117-008-2200-4