Let $H$ be a Hilbert space and $A$ be a $*$-subalgebra of $B(H)$ and $\mathbf 1\in A.$ Let $U$ denotes the closed unit ball of $A$. Given that $U$ is weakly closed. Then I want to show that $A$ is weakly closed.
I know that, weak topology and ultraweak topology coincides on unit ball and then I stuck here and do not know how to proceed. Please help me to solve this. Thank you for your help and time.
2026-04-14 05:18:49.1776143929
Unital $*$-subalgebra of $B(H)$ is weakly closed if its closed unit ball is weakly closed
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