On domains of unbounded derivations of generalized \({B}^{*}\)-algebras (Q1790416)

There are many results dealing with unbounded $^*$-derivations on $C^{*}$-algebras. For example, if $A$ is a $C^{*}$-algebra and $\delta : D(\delta) \rightarrow A$ is a closed unbounded $^*$-derivation of $A$, then $f(x) \in D(\delta)$ whenever $x \in D(\delta)$ and $f$ is an analytic function on a neighborhood of $\text{Sp}_{A}(x)$. In this case, the domain $D(\delta)$ is said to be closed under analytic functional calculus. \par This paper extends some of the known results of this type to generalizad $B^{*}$-algebras, $^*$-algebras of unbounded linear operators on a Hilbert space. As an example of one of the many results of the paper, the authors prove that, if $A$ is a complete $GB^{*}$-algebra with jointly continuous multiplication and $\delta : D(\delta) \rightarrow A$ is a closed $^*$-derivation of $A$ such that $1 \in D(\delta)$ and satisfies that $\sigma_{A}(x)=\sigma_{D(\delta)}(x)$ for all normal elements $x \in D(\delta)$, then $D(\delta)$ is closed under analytic functional calculus. After some preliminaries, the three main sections of the paper are devoted to finding sufficient conditions for the domain of a closed $^*$-derivation of a complete $GB^{*}$-algebra with jointly continuous multiplication to be closed under analytic functional calculus; providing examples and counterexamples on pro-$C^{*}$-algebras; finally, finding conditions under which a closed $^*$-derivation generates a one-parameter group of $^*$-automorphisms.