Let $H$ be an Hilbert space, and $A$ a von Neumann subalgebra of $B(H)$.
It is made abundantly clear that this is equivalent to: $A$ is a s-o closed *-subalgebra, and also $A$ is w-o closed *-subalgebra.
What is not clear (to me), is whether the actual occurences of convergence are the same, namely:
letting $a_i, a\in A$, does
$a_i \longrightarrow a$ (weakly) imply
$a_i \longrightarrow a$ (strongly).
Is this even true in when restricting attention to the unit ball of $A$?