Proposition 5

For any $x$ in A, let $t_x$ be the linear form $y\mapsto$ Trd$_{A/K}(xy)$ on A.

a) The mapping $t:x\mapsto t_x$ is an isomorphism of $(A,A)$-bimodules from A to its dual Hom$_K(A,K)$.

b) Let $h$ be a linear form on A. The following properties are equivalent:

(i) There exists an element $\lambda$ of K such that $h(x) =\lambda$ Trd$_{A/K}(x)$

for every $x\in A$.

(ii) We have $h(xy) =h(yx)$ for all $x, y$ in A.

Recall (II, §1, No. 14, p. 225–226) that the $(A$, A)-bimodule structure on $A^*=$ Hom$_K(A,K)$ is defined by the relation

for $a, b, c\in A$ and $t\in A^*$. In particular, for every $x$ in A, we have

$\langle at_xb, c\rangle =\langle t_x, bca\rangle =$ Trd$_{A/K}(xbca)$,

$\langle t_{axb}, c\rangle =$ Trd$_{A/K}(axbc)$,

and these two elements are equal by formula (37) of VIII, p. 342. We therefore have $at_xb=t_{axb}$, which means that $t$ is a homomorphism of $(A$, A)-bimodules from A to $A^*$.

We choose an extension L of the field K and an isomorphism $\theta$ from the L-algebra $A_{(L)}$ to the matrix algebra $\mathbf{M}_n(L)$ (VIII, p. 252, Theorem 1). We identify the vector space $(A^*)_{(L)}$ with the dual of the vector space $A_{(L)}$ over the field L. By Proposition 4 of VIII, p. 341, with these conventions, we have

(44) Trd$_{A_{(L)}/L}= 1_L\otimes$ Trd$_{A/K}$.

Let $t_{(L)}$ be the L-linear mapping from $A_{(L)}$ to $A^*_{(L)}$ deduced from $t$ by extension of scalars; by formula (44) and Corollary 1, we have

(45) $\langle t_{(L)}(x), y\rangle =$ Trd$_{A_{(L)}/L}(xy) =$ Tr($\theta (x)\theta (y)$)

for $x, y$ in $A_{(L)}$. By Proposition 7 of II, §10, No. 11, p. 358, the mapping $t_{(L)}$ is bijective; it follows that $t$ is bijective. We have proved a).

Let $h$ be in $A^*$; by the above, there exists an element $a$ of A such that $h$ is equal to $t_a$. By a), we have

Consequently, the relation “$h(xy) =h(yx)$ for $x, y$ in A” is equivalent to “$t_{ax-xa}= 0$ for every $x\in$ A”, and by part a) of the proof, this means that $a$ belongs to the center K of A. This proves b) in view of formula (36).