Let $A \subset \mathcal{B}(H)$ a subalgebra, not necessarily a $*$-algebra. In Murphy's book 'C*-algebras and Operator Theory', in Remark 4.2.1 you can find a proof of the failure of strong compactness for the ball of $\mathcal{B}(H)$:
If the ball is strongly compact, then the identity map of the ball with the relative strong topology to the ball with the relative weak operator topology is a continuous bijection from a compact space to a Hausdorff one, and therefore a homeomorphism, so the relative strong operator and the relative weak operator topologies coincide. But the involution operator is weakly continuous but not strongly continuous restricted to the ball, and it shows that the unit ball can't be strongly continuous.
It is clear that this proof relies in the fact that the two topologies can't coincide in the unit ball because the non-strong continuity of the involution operator, and it is clearly an extendable method to the $*$-subalgebras setting. But, if we have a subalgebra $A$, not necessarily closed by the adjoint map, are there any 'invariants' of the relative topologies that we can analyze to decide if the two topologies cannot coincide in the ball of $A$?
Thank you very much.