Spectral operation in locally convex algebras

Abstract

We show that if A is a spectrally bounded algebra, then all functions operate spectrally on A if and only if SpAx is finite for every x ∈ A. We also prove that if A is a commutative Q-l.m.c.a, then all functions operate spectrally on A if and only if A/RadA is algebraic. Furthermore, if A is a semi-simple commutative Q-l.m.c.a. which is a Baire space, all functions operate spectrally on A if and only if it is isomorphic to Cn for some n ∈ N. A structure result concerning semi-simple commutative complete m-convex algebras of countable dimension is also given.