Let $H$ be a separable Hilbert space. Let $A \subseteq B(H)$ be a finitely generated unital algebra. Let $M$ be the strong operator topology closure of $A.$
Let $B\_A$ be the closed ball in $A$ and $B\_M$ be the closed ball in $M.$
Is $B\_M$ the strong operator topology closure of $B\_A?$ (More likely, is there a counterexample?)



The ball formulation of the Kaplansky density theorem in nonselfadjoint algebras
W. R. Wogen, Some counterexamples in nonselfadjoint operator algebras, Ann. Math. 126 (1987), 415-427, constructs an operator $T$ for which the WOT and weak\* closures of the algebra generated by $T$ and $I$ are distinct. Since the WOT closure equals the SOT closure (and WOT and weak\* agree on bounded sets), this gives a counterexample to your version.


