Algebras of Hilbert Space Operators

Abstract

Collections of Hilbert space operators lead to a rich palette of algebraic structures. Of principal interest in this chapter are certain associative algebras of operators, called ∗-algebras, that are closed under the involution T ↦ T∗. However, one could also quite readily consider other algebraic structures, such as semigroups of operators, Lie algebras of operators, or vector spaces of operators. Our focus on ∗-algebras of Hilbert space operators (and their abstractions known as C∗-algebras) stems from the fact that such algebras are widely employed and studied, and exhibit special features that are not present in more generic algebraic structures. The monograph of Paulsen [42] has a good treatment of the theory of general operator algebras and discusses a wide variety of applications to operator theory.