Derivations in generalized $B^*$-algebras

We will be talking about derivations in the realm of generalized $B^*$-algebras ($GB^*$-algebras for short). The latter are algebras which densely contain a $C^*$-algebra but they may have unbounded operators adjoined. $GB^*$-algebras were first introduced by G. R. Allan and later studied by P. G. Dixon. Subsequently other researchers (among them Fragoulopoulou, Inoue, Kursten, Trapani et al.) found results on the structure of $GB^*$-algebras either by focusing their research interest directly on $GB^*$-algebras, or as by-products via their research on different algebraic ambients such as locally convex quasi $*$-algebras and $C^*$-like convex algebras.
After giving the main definitions and basic results on $GB^*$-algebras, we will discuss questions pertaining to continuity, innerness of derivations of $GB^*$-algebras, uniqueness of the zero derivation for a commutative $GB^*$-algebra, invariance of the domain of an everywhere defined derivation under analytic functional calculus as well as others. In doing so, we will observe in which extent these results go in parallel with the analogue well-known results in the $C^*$-algebra setting and in which instances they may differ.
The results presented in the talk are joint work with M. Weigt.