Remark 24

Suppose that the conditions of No. 3 are fulfilled. Let $\Gamma$ be a group of automorphisms of E such that every $u \in \Gamma$ is normal. Now prove that there exists a decomposition of E into direct sum of subspaces $E_k$ ($1 \leq k \leq r$) pairwise orthogonal, such that the restriction to $E_k$ of every $u \in \Gamma$ is of the form $\lambda_k \nu_k$, where $\lambda_k$ is an element $> 0$ of K and where $\nu_k$ is a unitary endomorphism of $E_k$ (decompose each $u \in \Gamma$ in the form $h \nu$, where $\nu$ is unitary, h positive hermitian and $h^2 = uu^*$ (exerc. 14 a)); use exerc. 23 b) and apply cor. 2 of th. 2 to the algebra generated by the hermitian endomorphisms h corresponding to the $u \in \Gamma$. Deduce that if $\Gamma$ is finite, the elements $u \in \Gamma$ are unitary.