Spectral Properties
Jan Okniński · 2020
This chapter considers some auxiliary “local” properties of elements of semigroup algebra. It exploits the obvious fact that the cardinality of the spectrum of a given element does not grow under homomorphism to show that spectrally bounded algebras over infinite fields are exactly semilocal algebras. The chapter proves that the class of “spectrally finite” algebras does not behave well under natural algebraic constructions by allowing the cardinality restriction on an infinite field to be overcome. It provides results on spectrally nondegenerated tensor products, as well as a theorem to prove that the algebra of a field of characteristic zero is a spectrally finite algebra if and only if a monoid is a locally finite semigroup.