DenseQ-subalgebras of Banach andC*-algebras and unbounded derivations of Banach andC*-algebras

Abstract

The paper studies denseQ-subalgebras of Banach andC*-algebras. It proves that the domainD(δ) of a closed unbounded derivation δ of a Banach unital algebraAautomatically contains the identity and is aQ-subalgebra ofA, so thatSpA(x) =SpD(δ)(x) for allx∈D(δ). The paper shows that every finite-dimensional semisimple representation of aQ-subalgebra is continuous. It also shows that if π is an injective *-homomorphism of a dense locally normalQ*-subalgebraBof aC*-algebra, then ‖x‖≦‖π(x)‖ for allx∈B. The paper studies the link between closed ideals of a Banach algebraAand of its dense subalgebraB. In particular, ifAis aC*-algebra andBis a locally normal *-subalgebra ofA, thenI→I∩Bis a one-to-one mapping of the set of all closed two-sided ideals inAonto the set of all closed two-sided ideals inBand.