Exercise 5, § 5

Let $E$ be a complex Hilbert space.

a) Let $n$ be an integer with $n\geqslant 2$. We denote by $I =\mathbf{N}\times \{0, . . . , n\}$. Let $(\sigma_i)_{1\leqslant i\leqslant n}$ be the family of permutations of I defined by

$\sigma_i(m, i) = (m-1, i)$ for $m\geqslant 1$

$\sigma_i(m, i-1) = (m+ 1, i-1)$ for $m\geqslant 0$

$\sigma_i(m, j) = (m, j)$ for $m\geqslant 0$ and $j \notin \{i, i-1\}$.

b) We denote by $(e_{m,i})$ the canonical orthonormal basis of $\ell^2(I)$. Let $u_i$ be the endomorphism of $\ell^2(I)$ such that $u_i(e_{m,j}) =e_{\sigma_i(m,j)}$ for all $(m, j)\in I$. The endomorphism $u_i$ is unitary.

c) The elements $\pi (u_i)$ are pairwise commuting.

d) There does not exist a family $(v_1, . . . , v_n)$ of unitary endomorphisms of $E$, pairwise commuting, such that $\pi (v_i) =\pi (u_i)$ for all $i$.