Lemma 10

Let L be a $(K,G)$-algebra that is étale over K. The mapping $\psi$ from $L_{(\Omega )}$ to $\mathscr{F}(\mathscr{H},\Omega )$ characterized by the relation

is an isomorphism of $(K,G)$-algebras.

Since L is étale, the mapping $\psi$ is an isomorphism of Ω-algebras (V, §6, No. 3, p. 30, Proposition 2 and V, §6, No. 3, p. 29, Proposition 1, c)). We have the relations

$\psi$((Id $\otimes \lambda (g)$)$(\xi \otimes x)$) $= (\xi (\chi \circ \lambda (g))(x))_{\chi\in\mathscr{H}}$

for $\xi \in \Omega ,x\in L$, and $g\in G$. So $\psi$ is a morphism of $(\Omega$, G)-algebras.