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.