Example

— *Let $H$ be a hilbertian space satisfying the first axiom of countability; let $H_{\sigma}$ denote the space $H$ with the weakened topology assigned to it. Let $\mathscr{L}^1(H)$ be the Banach space of nuclear endomorphisms of $H$ (V, p. 51, and TS, V); the norm in $\mathscr{L}^1(H)$ is defined by $\|u\|_1 = \operatorname{Tr}((u^*u)^{1/2})$. We can identify $\mathscr{L}(H)$ with the dual of the Banach space $\mathscr{L}^1(H)$ by associating the linear form $\phi_u : v \mapsto \operatorname{Tr}(uv)$ on $\mathscr{L}^1(H)$ with every $u \in \mathscr{L}(H)$. Let $A$ be a sub-algebra of $\mathscr{L}(H)$, containing 1 and stable under $u \mapsto u^*$; this is a von Neumann algebra if and only if it is closed in $\mathscr{L}(H)$ for the weak topology $\sigma(\mathscr{L}(H), \mathscr{L}^1(H))$. From cor. 3, we deduce the following criterion : for $A$ to be a von Neumann algebra, it is necessary and sufficient that if $(u_n)$ is any sequence of elements of $A$ with norm $\leqslant 1$ having a limit $u$ in the space $\mathscr{L}_s(H ; H_{\sigma})$, then $u$ belongs to $A$.