Generalized $B^*$-algebras: An introduction with applications to mathematical physics

Generalized $B^*$-algebras ($GB^*$-algebras, for short) are locally convex $*$-algebras which are abstract $*$-algebras of unbounded linear operators on a Hilbert space, and are generalizations of $C^*$-algebras. They were first studied by G. R. Allan in the late sixties, and then later, in the early seventies, by P. G. Dixon to include non locally convex algebras. The aim of this talk is to give an introduction and survey of these algebras, and we include some of the latest results of the speaker and his collaborators.
Observables in quantum mechanics are self-adjoint (unbounded) linear operators on a Hilbert space, and $GB^*$-algebras can be realized as $*$-algebras of unbounded linear operators on some Hilbert space. This talk will end with applications of $GB^*$-algebras to quantum entanglement and quantum irreversible dynamics.