Self-adjoint

In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e.

An element

is called self-adjoint if

[1] The set of self-adjoint elements is referred to as

{\displaystyle {\mathcal {A}}_{sa}}

A subset

that is closed under the involution *, i.e.

, is called self-adjoint.

[2] A special case of particular importance is the case where

is a complete normed *-algebra, that satisfies the C*-identity (

), which is called a C*-algebra.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian.

for the set of self-adjoint elements are also sometimes used, even in the more recent literature.