I was reading about the weak operator topology in wiki, and I saw that: Norm-bounded sets in $B(H)$ are pre-compact in WOT.
I was wondering: Is there an analogous statement for von Neumann algebras? $C^*$-algebras?
2026-03-31 01:02:14.1774918934
Norm bounded sets in $C^*$-algebras/von Neumann algebras
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The statement would have been that norm bounded sets are pre-compact, as being bounded does not guarantee that it is closed.
Other than that, yes. You would usually see a von Neumann algebra $M$ as embedded in $B(H)$. So a wot-closed bounded set is a closed subset of a compact set (a closed ball in $B(H)$) and thus compact.
In the case of a C$^*$-algebra, you would have to embed it in some $B(H)$ to even have a wot topolgy. In that case, bounded sets would be pre-compact.