Spectral operation in locally convex algebras
DOI:
https://doi.org/10.22199/issn.0717-6279-4548Keywords:
spectrally bounded algebra, Q-algebra, l.m.c.a, algebraic algebra, Baire space, Fourier-Gelfand transform, algebra of countable Hamel basis, operate function, function operate spectrallyAbstract
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.
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