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Systems of equations in the predual of a von Neumann algebra

Author: Michael Marsalli
Journal: Proc. Amer. Math. Soc. 111 (1991), 517-522
MSC: Primary 46L10; Secondary 47A62, 47D27
MathSciNet review: 1042269
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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}$.

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Keywords: Von Neumann algebra, dual algebra, predual
Article copyright: © Copyright 1991 American Mathematical Society

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