The finest topology that coincides with $\tau*$ ($\sigma$-strong in this case) on $\tau$-bounded (norm-bounded in this case) subsets is the mixed topology $\gamma(\tau,\tau^*)$, introduced by [A. Wiweger, Linear spaces with mixed topology. Studia Mathematica 20 (1961), 47--68][1]; see 2.2.2. For the Hilbert space $\ell_2$ (or perhaps any separable infinite dimensional Banach space?), the inequalities 
$$\mbox{$\sigma$-strong} \le \mbox{uniform convergence on compact subsets} \le \gamma(\mbox{norm},\mbox{$\sigma$-strong})$$
are strict (Addendum: I'm no longer so sure if the second inequality is strict). 

  [1]: https://eudml.org/doc/217004

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A note for posterity: I was noticed that there is some confusion in the literature as to the difference of the above topologies. E.g., Proposition I.8.6.3 in [Blackadar's book][2] erroneously claims that the $\sigma$-strong topology on $B(H)$ coincides with the strict topology arising as the multiplier of $K(H)$. The essentially same claim is seen in Chapter 8 in [Lance's book][3]. In fact, these topologies are different (although they coincide on norm bounded subsets, and that's what's needed in daily life), as the following example shows. 

Take a sequence of mutually orthogonal rank one orthogonal projections $p_n$ on a separable Hilbert space $H$. 
Then, the $\sigma$-strong closure of $\{ \sqrt{n} p_n \}$ contains $0$. Indeed, for any positive linear functional $f$ on $B(H)$, one has $\liminf_n f(n p_n)=0$, for otherwise $f(p_n)>\frac{\epsilon}{n}$ for all $n$ and $\lim_m f(\sum_{n=1}^m p_n)=\infty$, a contradiction. However, $T:=\sum_{n=1}^\infty \frac{1}{\sqrt{n}}p_n \in K(H)$ (or the compact subset $\{ Tv : \|v\|\le1\})$ separates $0$ from $\{ \sqrt{n} p_n \}$ in the strict topology (or the compact open topology). 

  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=2188261
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=1325694