Quasi $^*$-algebras and generalized inductive limits of $C^*$-algebras
Volume 202 / 2011
Studia Mathematica 202 (2011), 165-190
MSC: Primary 47L60; Secondary 47L40.
DOI: 10.4064/sm202-2-4
Abstract
A generalized procedure for the construction of the inductive limit of a family of $C^*$-algebras is proposed. The outcome is no more a $C^*$-algebra but, under certain assumptions, a locally convex quasi $^*$-algebra, named a $C^*$-inductive quasi $^*$-algebra. The properties of positive functionals and representations of $C^*$-inductive quasi $^*$-algebras are investigated, in close connection with the corresponding properties of positive functionals and representations of the $C^*$-algebras that generate the structure. The typical example of the quasi $^*$-algebra of operators acting on a rigged Hilbert space is analyzed in detail.
