Definition 1

Let A be an algebra over $\mathbf{C}$. An involution in A is called a mapping $x\mapsto x^*$ of A into A such that:

for all $x, y\in A$ and $\lambda \in \mathbf{C}$. An algebra over $\mathbf{C}$ endowed with an involution is called an involutive algebra.

An involution on A is in particular an isomorphism of the ring A onto the opposite ring $A^{\circ}$.

Let A be an involutive algebra. One says that $x^*$ is the adjoint of $x$. A subset of A stable under the involution is said to be self-adjoint. If A possesses a unit element $e$, one has $e^*=e$; one says that $(A, e)$ is a unital involutive algebra. An element $u$ of a unital involutive algebra is said to be unitary if $uu^*=u^*u=e$, in other words if $u$ is invertible and if its inverse is $u^*$.

An element $x\in A$ is said to be hermitian if $x=x^*$ and normal if $xx^*=$ $x^*x$. This terminology generalizes that of A, IX, § 7, No.$^o3$. Every hermitian element is normal, every unitary element is normal. The set $A_h$ of hermitian elements of A is a real vector subspace of A. If $x$ and $y$ are hermitian and permutable, one has $(xy)^*$ = $y^*x^*=yx=xy$, hence $xy$ is hermitian. For every $x\in A$, the elements $xx^*$ and $x^*x$ of A are hermitian.

If $A =\mathbf{C}$ endowed with the involution $z\mapsto \overline{z}$, one has $A_h=\mathbf{R}$.