Systems of equations in the predual of a von Neumann algebra

Abstract:

A von Neumann algebra $\mathcal {A}$ on a separable, complex Hilbert space $\mathcal {H}$ has property ${{\mathbf {A}}_n}$ if for every $n \times n$ array $\{ {f_{i,j}}\}$ of elements in the predual there exists sequences $\{ {x_i}\} ,\{ {y_j}\}$ in $\mathcal {H}$ such that ${f_{i,j}}(A) = (A{x_i},{y_j})$ for all $A$ in $\mathcal {A}$ and $0 \leq i,j < n$. We show that the von Neumann algebras with property ${{\mathbf {A}}_{{\aleph _0}}}$ are the von Neumann algebras with properly infinite commutant. We describe how these properties are transformed by the tensor product. We characterize the abelian von Neumann algebras with property ${{\mathbf {A}}_n}$.