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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]; 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.