On domains of unbounded derivations of generalized B $^{*}$ -algebras

Abstract

We study properties under which the domain of a closed derivation δ:D(δ)→A of a generalized B∗-algebra A remains invariant under analytic functional calculus. For a complete, generalized B∗-algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of A belongs to the domain of the derivation, along with a condition related to the coincidence σA(x)=σD(δ)(x) of the (Allan) spectra for every element x∈D(δ). Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed ∗-derivation δ of a complete GB∗-algebra with jointly continuous multiplication such that 1∈D(δ) and x a normal element of the domain, f(x)∈D(δ) for every analytic function on a neighborhood of the spectrum of x. We also give an example of a closed derivation of a GB∗-algebra which does not contain the identity element. A condition for a closed derivation of a GB∗-algebra A to be the generator of a one-parameter group of automorphisms of A is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.