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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 $A$?

2 Answers 2