Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1991-1042269-3
MathSciNet review: 1042269
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.


References [Enhancements On Off] (What's this?)

  • [1] Hari Bercovici, Ciprian Foias, and Carl Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 787041
  • [2] Scott W. Brown, Some invariant subspaces for subnormal operators, Integral Equations Operator Theory 1 (1978), no. 3, 310–333. MR 511974, https://doi.org/10.1007/BF01682842
  • [3] Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
  • [4] A. Loginov and V. Sulman, Hereditary and intermediate reflexivity of $ {W^ * }$-algebras, Math. USSR-Izv. 9 (1975), 1189-1201.
  • [5] F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, https://doi.org/10.2307/1968693
  • [6] D. Topping, Lectures on von Neumann algebras, Van Nostrand Rheinhold, London, 1971.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L10, 47A62, 47D27

Retrieve articles in all journals with MSC: 46L10, 47A62, 47D27


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1042269-3
Keywords: Von Neumann algebra, dual algebra, predual
Article copyright: © Copyright 1991 American Mathematical Society