Affiliated subspaces and infiniteness of von Neumann algebras

作者：Hamhalter Jan^*; Turilova Ekaterina

来源：Mathematische Nachrichten, 2013, 286(10): 976-985.

DOI：10.1002/mana.201200157

• Subspaces affiliated with a von Neuman algebra
• GNS representation
• subspaces in pre-Hilbert spaces
• quasi-splitting subspaces MSC (2010) 46L10
• 46L30
• 46C05

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi-splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi-splitting subspaces are non-equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi-splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.

• 出版日期2013-7