Exercise 6, § 5

Let E be a complex Hilbert space. Recall that $\mathscr{C}$alk(E) is a star algebra (exercise 3).

Let $\mathfrak{F}$ be a non-principal ultrafilter on $\mathbf{N}$. Let F be the vector space of bounded sequences in E which converge weakly to 0 according to $\mathfrak{F}$ and $F_0$ the subspace of F formed by the sequences which converge in norm to 0 according to $\mathfrak{F}$. We put $\widetilde{H} = F/F_0$. a) For all elements $x= (x_n)$ and $y= (y_n)$ of F, now prove that the sequence $(\langle x_n|y_n\rangle )_{n\in\mathbf{N}}$ converges according to $\mathfrak{F}$. We denote by $\langle x|y\rangle_F$ its limit.

b) Now prove that $(x, y)\mapsto \langle x|y\rangle_F$ is a positive Hermitian form on F, and that $\langle x|x\rangle_F= 0$ if and only if $x\in F_0$; this Hermitian form therefore defines on $\widetilde{H}$ a structure of separated prehilbert space.

c) Let H be the completed Hilbert space of the separated prehilbert space $\widetilde{H}$. Now prove that for every $u$ in $\mathscr{L}$ (E), one obtains an element $\varrho (u)$ of $\mathscr{L}(H)$ by passing to the quotient from the linear mapping $(x_n)\mapsto (u(x_n))$ of F. d) The mapping $u\mapsto \varrho (u)$ induces an isometric morphism of star algebras $\mathscr{C}$alk$(E)\rightarrow \mathscr{L}(H)$.

(We shall see further on that every star algebra A possesses an isometric representation $A\rightarrow \mathscr{L}(E)$ for a certain Hilbert space E, cf. th. 2 of V, p. 442.)