Banach partial $*$-algebras: an overview

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