Exercise 15, § 4

Let E be a complex Hilbert space; denote by $\mathscr{B} = \mathscr{L}(E)$ the algebra of continuous endomorphisms of E, endowed with the usual norm $\|T\| = \sup_{\|x\| \leq 1} \|T.x\|$. For every pair of points x, y of E, denote by $\omega_{x,y}$ the continuous linear form $T \mapsto \langle x|T.y \rangle$ on $\mathscr{B}$, and denote by $\mathscr{B}_o$ the closed subspace of the strong dual $\mathscr{B}'$ of the Banach space $\mathscr{B}$, generated by the $\omega_{x,y}$. a) Show that the linear mapping which, to every $T \in \mathscr{B}$, associates the linear form $\omega \mapsto \langle \omega, T \rangle$ on $\mathscr{B}_o$, is an isometry of $\mathscr{B}$ onto the strong dual of $\mathscr{B}_o$; in other words, $\mathscr{B}_o$ is a preDual (IV, p. 56, exerc. 23) of $\mathscr{B}$. The topology $\sigma(\mathscr{B}, \mathscr{B}_o)$ on $\mathscr{B}$ is called the ultraweak topology. b) For every element $T$ of $\mathscr{L}^1(E)$, define a linear form $\varphi_T$ on $\mathscr{B}$ by the formula $\varphi_T(S) = \operatorname{Tr}(ST)$ for every operator $S \in \mathscr{B}$. Show that $\varphi_T$ is continuous, and that the mapping $T \mapsto \varphi_T$ is an isometry of the Banach space $\mathscr{L}^1(E)$ (exerc. 14, e)) onto the Banach space $\mathscr{B}_o$ (consider first the case where T is of finite rank). c) Let $\mathscr{B}_{oo}$ be the vector subspace of $\mathscr{B}_o$ generated by the $\omega_{x,y}$ (so that $\mathscr{B}_o = \overline{\mathscr{B}_{oo}}$). Show that $\mathscr{B}_{oo}$ is barrelled (observe that a subset of $\mathscr{B}$ bounded for $\sigma(\mathscr{B}, \mathscr{B}_{oo})$ is bounded for the norm topology). d) Let $F_n$ be the subspace of $\mathscr{B}_{oo}$ image of the set of endomorphisms of E of finite rank $\leq n$ by the isometry defined in b). Show that $F_n$ is nowhere dense in $\mathscr{B}_{oo}$ and deduce that $\mathscr{B}_{oo}$ is not a Baire space.