Hermitian and algebraic $^*$-algebras, representable extensions of positive functionals
Volume 256 / 2021
Studia Mathematica 256 (2021), 311-343
MSC: Primary 46L30, 46H10; Secondary 43A20, 46L05, 46L35, 47L40.
DOI: 10.4064/sm190722-11-12
Published online: 2 July 2020
Abstract
We introduce the concept of $E^*$-algebras in the context of representable extensions of positive functionals, and we show that the class of $E^*$-algebras coincides with the class of hermitian algebraic $^*$-algebras. Results on Banach $E^*$-algebras and pre-$C^*$-algebras which are $E^*$-algebras are also included.
As applications we give a characterization of discrete locally finite groups via $E^*$-algebras and answer Kurosh’s problem in the affirmative in the case of the convolution $^*$-algebras of compactly supported continuous functions on an arbitrary locally compact group.
