Rickart ∗-Rings, Baer ∗-Rings, AW*-algebras: Generalities and Examples

Abstract

All rings considered in this book are associative, and, except in a few of the excercises, they are equipped with an involution in the sense of the following definition:

Definition 1. A ∗-ring (or involutive ring, or ring with involution) is a ring with an involution x↦x^∗:

$$ (x^*)^*=x,\quad (x+y)^*=x^*+y^*,\quad (xy)^*=y^*x^*.$$

When A is also an algebra, over a field with involution λ↦λ^∗ (the identity involution is allowed), we assume further that

$$(\lambda x)^*=\lambda^*x^*$$

and call A a ∗-algebra {The complex ∗-algebras are especially important special cases, but the main emphasis of the book is actually on ∗-rings.}