Proposition 34

Let $H$ be a Lie group, $A$ a unital complete normable associative algebra and $\phi : H \to A^*$ a Lie group morphism. The associated morphism $\phi'$ of $\mathscr{T}^{(\infty)}(H)$ into $A$ is obtained by composing $\phi_*$ with the canonical morphism of $\mathscr{T}^{(\infty)}(A^*)$ into $A$. In particular, $\phi'(x) = L(\phi)(x)$ for all $x \in L(H)$.

Let $i$ be the identity mapping of $A^*$ into $A$. Then, for all $t \in \mathscr{T}^{(\infty)}(H)$,