Let $M$ be a von Neumann algebra. Let $x,y$ be two self-adjoint operators in $M$. Are there any von Neumann subalgebra $A$ of $M$ containing $x,y$ such that the predual of $A$ is separable?
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Not necessarily. In fact, even a single operator $x$ can generate a nonseparable vNa. For example, in the sequence space $l^2([0, 1])$, the diagonal operator given by $x(e_t) = te_t$ for all $t \in [0, 1]$ generates $l^\infty([0, 1])$, which is not separable. Though your question will have a positive answer if you assume $M$ is countably decomposable, since a countably generated and countably decomposable vNa is separable. (This can be seen by considering a GNS construction associated with a normal faithful state. Then countable generation implies the Hilbert space is separable.)
