Let $\mathcal{M}$ be a $W^*$-algebra, i.e. a $C^*$-algebra with a Banach space predual $\mathcal{M}_*$. I'm trying to show that the adjoint map $x \mapsto x^*$ on $\mathcal{M}$ is weak-* (aka $\sigma$-weakly) continuous.
Chapter 1.7 of Sakai's $C^*$-Algebra and $W^*$-Algebras starts by proving that the real-linear subspace $\mathcal{M}^s$ of self-adjoint elements is weak-* closed. Later (in 1.7.2 as well as 1.7.8) he asserts that this, plus the fact that $\mathcal{M}^s \cap i \mathcal{M}^s = \{0\}$ and $\mathcal{M}^s + i \mathcal{M}^s = \mathcal{M}$, imply that the adjoint map is weak-* continuous. I'm afraid I don't quite follow.
- We know that $\mathcal{M}$ is the algebraic direct sum of the weak-* closed, real-linear subspaces $\mathcal{M}^s$ and $i \mathcal{M}^s$. However, not every algebraic complement is a topological complement. I don't know of many sufficient conditions for algebraic complements to automatically be topological. One is that the space in question be Fréchet, but if $\mathcal{M}_*$ is infinite-dimensional then $\mathcal{M}$ cannot be weak-* Fréchet.
- The restriction of the adjoint map to the unit ball of $\mathcal{M}$ is continuous: If $x_\nu + i y_\nu \to x+iy$ is a convergent net in the unit ball with $x,y,x_\nu, y_\nu$ self-adjoint, then $x_\nu, y_\nu, x, y$ are also in the unit ball; given any subnet, there exists (by Alaoglu) a sub-subnet for which $x_\nu$ converges weak-* to some $\tilde{x}$ in the unit ball, and a sub-sub-subnet for which $y_\nu$ also converges weak-* to some $\tilde{y}$ in the unit ball. (I'm using the same notation for all subnets instead of writing things like $x_{\nu_{\mu_{\eta_\zeta}}}$.) Because $\mathcal{M}^s$ is weak-* closed, it follows that $\tilde{x}$ and $\tilde{y}$ are self-adjoint, and since the sub-sub-subnet $x_\nu + i y_\nu$ converges to both $x+iy$ and $\tilde{x} + i\tilde{y}$, it follows that $\tilde{x} = x$ and $\tilde{y} = y$. Then (for this same sub-sub-subnet) one has $x_\nu - i y_\nu \to x-iy$. Since every subnet of $x_\nu - i y_\nu$ has a further subnet converging to $x-iy$, we have $x_\nu - i y_\nu \to x-iy$.
- Not sure how to finish given the above remarks on the unit ball. Given a weak-* convergent net $m_\nu \to 0$, the above would immediately imply $m_\nu^* \to 0$ weak-* as well if we made the additional assumption that the net $m_\nu$ is eventually bounded...but not all weak-* convergent nets are. Or, for any weak-* closed convex set $F \subset \mathcal{M}$, let $F_r$ denote the intersection of $F$ with the ball of radius $r$, and we then get that $F_r^*$ is weak-* closed; by Krein-Smulyan, it follows that $F^*$ is weak-* closed. However, most closed sets aren't convex.
Of course, one approach would be to (without using continuity of the adjoint) develop the theory of $W^*$-algebras far enough to get a representation theorem, then use the ultraweak continuity of the adjoint map on $B(H)$. I'd really rather have something more direct, though! I have a feeling I'm overlooking something obvious.