On domains of unbounded derivations of generalized B∗-algebras

Abstract

We study properties under which the domain of a closed derivation $\delta :D\left(\delta \right)\to A$ of a generalized B${}^{\ast }$-algebra $A$ remains invariant under analytic functional calculus. For a complete, generalized B${}^{\ast }$-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 ${\sigma }_A\left(x\right)={\sigma }_{D\left(\delta \right)}\left(x\right)$ of the (Allan) spectra for every element $x\in D\left(\delta \right)$. 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 $\ast$-derivation $\delta$ of a complete GB${}^{\ast }$-algebra with jointly continuous multiplication such that $1\in D\left(\delta \right)$ and $x$ a normal element of the domain, $f\left(x\right)\in D\left(\delta \right)$ for every analytic function on a neighborhood of the spectrum of $x$. We also give an example of a closed derivation of a GB${}^{\ast }$-algebra which does not contain the identity element. A condition for a closed derivation of a GB${}^{\ast }$-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.