Exercise 2, § 19

a) Let A be a simple ring of finite rank over its center K, and let $s$ be an involutive antiautomorphism of A. Let $s'$ be an involutive antiautomorphism of A that coincides with $s$ on K. Prove that there exists an element $a$ of A such that we have $s'(x) =as(x)a^{-1}$ for every $x\in A$ and $s(a) =a$ or $s(a) =-a$ (apply the Skolem–Noether theorem and Theorem 3 of V, §11, No. 6, p. 85). Also prove the converse.

b) Suppose that the restriction of $s$ to K is not the identity; let $K_0$ be the subfield of K consisting of the elements fixed by $s$. Prove that the set $A_0$ (resp. $A_1$) of elements $a$ of A such that $s(a) =a$ (resp. $s(a) =-a$) is a $K_0$-linear subspace of A of dimension [A : K] and that $A_0$ is a $K_0$-structure on A (use V, §10, No. 4, p. 63, Proposition 7).

c) Suppose that the restriction of $s$ to K is the identity; set $[A : K] =m^2$. Prove that $A_0$ is a $K_0$-linear subspace of A of dimension $m(m+ 1)/2$ or $m(m-1)/2$ (reduce to the case when A is a matrix algebra over K and use a)).