Let $H$ be a complex Hilbert space, and let $\mathcal{B}(H)$ denote the algebra of bounded operators on $H$. It is known that the strong operator topology and the norm topology on $\mathcal{B}(H)$ coincide if and only if $H$ is finite dimensional. But the strong operator topology cannot be characterized using sequences, so we have to use nets. My question is: Can the strong and norm topologies coincide sequentially- id est, every strong convergent sequence is norm convergent - in a certain subalgebra $A \subseteq \mathcal{B}(H)$ with infinite dimension?

Thank you very much.