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 xL is Abel bounded and \(x^N \geq 1\) for some \(L,N \in \mathbb{N}\).
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Braatvedt, G., Brits, R. & Raubenheimer, H. Gelfand-Hille type theorems in ordered Banach algebras. Positivity 13, 39–50 (2009). https://doi.org/10.1007/s11117-008-2200-4
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DOI: https://doi.org/10.1007/s11117-008-2200-4