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Let $H$ be a Hilbert space, and $A$ a $C^*$-subalgebra of $B(H)$ (the bounded operators on $H$). Let $B$ be the strong-operator closure of $A$, so that in particular, $B$ is a von-Neumann-algebra.

According to the Kaplansky-Theorem:

The … in the unit ball of $A$ is s-o dense in the … in unit ball of $B$, where … is:

  • unitaries
  • self adjoints

Does it hold, that $\operatorname{Proj}(A)$ is strongly dense in $\operatorname{Proj}(B)$ (or even weakly dense).

I.e. does the Kaplansky-Theorem hold for projections? (Or a weaker version of it.)

If not, why not?

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