Banach partial $*$-algebras: an overview

A Banach partial *-algebra is a locally convex partial *-algebra whose total space is a Banach space. A Banach partial *-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of these objects and display a number of examples, namely, Lp-like function spaces and spaces of operators on Hilbert scales or lattices. Finally we analyze the important cases of Banach quasi *-algebras and CQ-algebras.