Definition 1

Let $\varepsilon$ be an element of the center of A. A sesquilinear form $\Phi$ on E such that one has $\Phi(x, y) = \varepsilon \overline{\Phi(y, x)}$ whatever $x$ and $y$ in E may be is called an $\varepsilon$-hermitian form. A 1-hermitian form (resp. (−1)-hermitian) is called hermitian (resp. antihermitian).

When J is the identity (which implies that A is commutative) a hermitian form (resp. antihermitian form) (for J) is nothing other than a symmetric (resp. antisymmetric) bilinear form (Chapter III, § 5, No. 1, def. 2). Recall that an alternating bilinear form (Chapter III, § 5, No. 2, def. 4) is antisymmetric; the converse is true if, in A, the relation $2a = 0$ implies $a = 0$.

The orthogonality relation ($§ 1, n° 3$) with respect to an $\varepsilon$-hermitian form is obviously symmetric (cf. exerc. 1).

If $\alpha$ is an invertible element of A, the mapping $T : \lambda \to \alpha^{-1} \overline{\lambda} \alpha$ is an antiautomorphism of A, and one readily verifies that the form $\Phi \alpha$ is sesquilinear with respect to T. If, moreover, one has $\alpha = \overline{\alpha}$, then T is involutive, and, if $\Phi$ is $\varepsilon$-hermitian, $\Phi \alpha$ is so also; indeed one has

In particular, when A is a field, the elements $\alpha$ of the center of A such that $\overline{\alpha} = \alpha$ form a subfield K of A, and the $\varepsilon$-hermitian forms on E (for J) form a vector space over K.