Bounded elements and spectrum in Banach quasi $^*$-algebras
Volume 172 / 2006
Studia Mathematica 172 (2006), 249-273
MSC: Primary 46L08; Secondary 46L51, 47L60.
DOI: 10.4064/sm172-3-4
Abstract
A normal Banach quasi $^*$-algebra $({\mathfrak X},)$ has a distinguished Banach $^*$-algebra ${\mathfrak X}_{\rm b}$ consisting of {bounded} elements of ${\mathfrak X}$. The latter $^*$-algebra is shown to coincide with the set of elements of ${\mathfrak X}$ having finite spectral radius. If the family ${\cal P}({\mathfrak X})$ of bounded invariant positive sesquilinear forms on ${\mathfrak X}$ contains sufficiently many elements then the Banach $^*$-algebra of bounded elements can be characterized via a $C^*$-seminorm defined by the elements of ${\cal P}({\mathfrak X})$.
