In mathematics, an element *x* of a star-algebra is **self-adjoint** if the involution acts trivially upon it. In other words, . When we work in an inner product space which is a star-algebra, being self-adjoint is the same as being hermitian.

A collection *C* of elements of a star-algebra is **self-adjoint** if it is closed under the involution operation. For example, if then since in a star-algebra, the set {*x*,*y*} is a self-adjoint set even though *x* and *y* need not be self-adjoint elements.

In linear algebra, an operator *T∈L(V)* is called **self-adjoint** iff *T = T**.

See also: