TITLE: The topology of pointwise convergence with the adjoint operator on a von Neumann algebra
QUESTION [6 upvotes]: Let $H$ be a Hilbert space and ${\mathcal A}$ a von Neumann algebra in $B(H)$. Let us endow ${\mathcal A}$ with the topology of pointwise convergence with the Hermitian adjoint operator, i.e. a net $A_i\in {\mathcal A}$ tends to 0 in ${\mathcal A}$ if
$$
\forall x\in H\qquad A_ix\overset{H}{\underset{i\to\infty}{\longrightarrow}}0 \quad\&\quad
A_i^*x\overset{H}{\underset{i\to\infty}{\longrightarrow}}0.
$$
The algebra ${\mathcal A}$ with this topology has totally bounded sets $S\subseteq {\mathcal A}$, which I believe, are described by the property
$$
\forall x\in H\qquad \text{the sets $\{Ax;\ A\in S\}$ and $\{A^*x;\ A\in S\}$ are totally bounded in $H$}
$$
Let $u:{\mathcal A}\to{\mathbb C}$ be a linear functional, which is continuous on every totally bounded set (equivalently, on each compact set) $S\subseteq {\mathcal A}$ in this sense (with respect to this topology induced from ${\mathcal A}$ to $S$).
Question:

does $u$ belong to ${\mathcal A}_*$ (the predual of ${\mathcal A}$)?

EDIT. I must say, even the case of ${\mathcal A}=B(H)$ is not clear for me. And even if we replace the topology on ${\mathcal A}$ with the usual strong operator topology (i.e. if we remove the requirement $A^*_ix\to 0$). We also can change premises in other different ways, for example, we can think that ${\mathcal A}$ is commutative (for me that would be sufficient now). If anybody could cast some light on any weakened conditions, I would appreciate this very much. I need this for my current work, this is related to the study of envelopes of topological algebras.

REPLY [2 votes]: This is a comment on the above answer but it will be too long and in any case I am not entitled.  The questions you pose are answered in much more detail in a Comptes Rendus article "Topologies sur les espaces des opérateurs dans l'espace hilbertien" (A276 (1973) 1509).  This discusses three locally convex topologies on a von Neumann algebra---the finest locally convex ones which coincide with the weak operator, the strong operator and the symmetric strong operator topology on the unit ball.  They are all complete, have the same convergent sequences as the above-mentioned ones, have the same dual space (the normal functionals) and have the property that a linear functional thereon is continuous if and only it is continuous on the unit ball for each or any of the above operator topologies.  This can be found in the secondary literature (in english) as chapter IV of "Saks spaces and applications to functional analysis" which is readily available online.  Particularly relevant is  Prop. 2.3 on p. 204. The commutative case is treated in Ch. 3.