Abstract
Let E be a field provided with an absolute value for which it is complete and let A be a commutative E-algebra with unity provided with a semimultiplicative (or power multiplicative) E-algebra semi-norm ‖.‖. Let Mult(A, ‖. ‖) be the set of multiplicative E-algebra semi-norms continuous with respect to ‖. ‖. We show the existence of a Shilov boundary for ‖. ‖, i.e. a closed subset F of Mult(A, ‖. ‖), minimal for inclusion, such that for every x∈A, there exists ϕ∈F such that ϕ(x)=‖x‖. In particular, if the field E is ultrametric, it applies to the spectral semi-norm of an ultrametric E-algebra.
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Escassut, A., Netti, N.M. (2004). Shilov Boundary for Normed Algebras. In: Barsegian, G.A., Begehr, H.G.W. (eds) Topics in Analysis and its Applications. NATO Science Series II: Mathematics, Physics and Chemistry, vol 147. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2128-3_1
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DOI: https://doi.org/10.1007/1-4020-2128-3_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-2062-9
Online ISBN: 978-1-4020-2128-2
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