Proposition 4

a) There exists a group homomorphism $\lambda_S$ from $S\otimes_DS^*$ to $\Theta_S(A)$, characterized by

for $x\in S$ and $x^*\in S^*$. This mapping is an isomorphism of $(A,A)$-bimodules.

b) The $(A,A)$-bimodule $\Theta_S(A)$ is simple.

The mapping $\gamma_S: S\otimes_KS^*\rightarrow \Theta_S(A)$ characterized by the formula $\gamma_S(x\otimes x^*) =c_S(x, x^*)$ is $(A$, A)-linear and surjective and satisfies $\gamma_S(xd\otimes x^*)$ $=\gamma_S(x\otimes dx^*)$ for $x\in S,x^*\in S^*$, and $d\in D$. It therefore defines a surjective $(A$, A)-linear mapping $\lambda_S: S\otimes_DS^*\rightarrow \Theta_S(A)$ characterized by formula (6).

Let us prove that $S\otimes_DS^*$ is a simple $(A$, A)-bimodule. By Corollary 2 of VIII, p. 63, every $(A$, A)-sub-bimodule of $S\otimes_DS^*$ is of the form $S\otimes_DH$, where H is a $(D$, A)-sub-bimodule of $S^*$. Since $S^*$ is a simple right A-module (VIII, p. 380, Theorem 1, c)), we have H = 0 or $H = S^*$; the assertion follows.

The homomorphism $\lambda_S$ is $(A$, A)-linear and nonzero; since $S\otimes_DS^*$ is a simple bimodule, $\lambda_S$ is injective (VIII, p. 47, Proposition 2, a)).