SPECTRAL ALGEBRAS

Spectral algebras are a class of abstract complex algebras which share many of the good properties of Banach algebras. In the commutative case they are precisely the class of abstract algebras having a full Gelfand theory. Any irreducible representation of a spectral algebra is strictly dense. Spectral algebras are defined and characterized in terms of spectral pseudo-norms and spectral subalgebras. Spectral algebras, spectral subalgebras and spectral pseudo-norms are shown to occur frequently in analysis. It is also shown that when the spectral radius is finite valued, if it is either subadditive or submultiplicative, then it has both properties and that this occurs exactly for algebras which are spectral algebras and commutative modulo their Jacobson radicals. The paper is written in an expository style.