Editing Self-adjoint (section)

== Definition ==

Let <math>\mathcal{A}</math> be a *-algebra. An element <math>a \in \mathcal{A}</math> is called self-adjoint if {{nowrap|<math>a = a^*</math>.{{sfn|Dixmier|1977|p=4}}}}

The [[Set (mathematics)|set]] of self-adjoint elements is referred to as {{nowrap|<math>\mathcal{A}_{sa}</math>.}}

A [[subset]] <math>\mathcal{B} \subseteq \mathcal{A}</math> that is [[Closed set|closed]] under the [[Involution (mathematics)|involution]] *, i.e. <math>\mathcal{B} = \mathcal{B}^*</math>, is called {{nowrap|self-adjoint.{{sfn|Dixmier|1977|p=3}}}}

A special case of particular importance is the case where <math>\mathcal{A}</math> is a [[Banach algebra#Banach *-algebras|complete normed *-algebra]], that satisfies the C*-identity (<math>\left\| a^*a \right\| = \left\| a \right\|^2 \ \forall a \in \mathcal{A}</math>), which is called a [[C*-algebra]].

Especially in the older literature on *-algebras and C*-algebras, such elements are often called {{nowrap|hermitian.{{sfn|Dixmier|1977|p=4}}}} Because of that the notations <math>\mathcal{A}_h</math>, <math>\mathcal{A}_H</math> or <math>H(\mathcal{A})</math> for the set of self-adjoint elements are also sometimes used, even in the more recent literature.