Banach partial $*$-algebras‎: ‎an overview

Document Type: Special issue: Trends in Operators on Banach Spaces


1 Université catholique de Louvain - IRMP

2 Dipartimento di Matematica e Informatica, Università di Palermo


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 $L^p$-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.