Let $M$ be a W*-algebra and consider the following map: $$\gamma: M\times M^*\to M^*: (a,f)\to af$$ where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under the weak-star topology $\sigma(M^*,M)$.

Q: Is $\gamma$ weakly-weak-star jointly continuous? I mean, assume that $\{a_i\}$ is weakly convergent to $a$ in $M$ and $\{f_i\}$ is weak-star convergent to $f$ in $M^*$. Then can we conclude that $a_if_i$ is weak-star convergent to $af$?

Naturally this question may be also asked for the bilinear mapping $M^{**}\times M^*\to M^*$.