Like the title: is it true that a self-adjoint unital subalgebra of $L(H)$ closed in the weak operator topology (a Von Neumann algebra) has a trivial null space? Why? Thank you.
2026-03-29 20:55:38.1774817738
Has a *subalgebra of $L(H)$ with $1$ a trivial null space?
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If the unit of the subalgebra is the identity operator (that is, the algebra is non-degenerate), then the answer is yes: if $A\xi=0$ for all $A$, then in particular $\xi=I\,\xi=0$.
If the unit of the subalgebra is a projection $P\ne I$, then the null space is $(I-P)L(H) $.