Doubly power-bounded elements with finite spectrum in Banach algebras admit a spectral decomposition extending Gelfand's theorem, with a generalization of Koehler-Rosenthal results and initial links to commutativity.
Zem´ anek, On the Gel’fand-Hille theorems, Functional analysis and operator theory (Warsaw, 1992), 369–385
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Spectral decomposition of doubly power-bounded elements in Banach algebras
Doubly power-bounded elements with finite spectrum in Banach algebras admit a spectral decomposition extending Gelfand's theorem, with a generalization of Koehler-Rosenthal results and initial links to commutativity.