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Affiliated subspaces and infiniteness of von Neumann algebras
Author(s) -
Hamhalter Jan,
Turilova Ekaterina
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200157
Subject(s) - linear subspace , mathematics , von neumann algebra , affiliated operator , von neumann architecture , subspace topology , abelian von neumann algebra , pure mathematics , algebra over a field , metric (unit) , order (exchange) , tomita–takesaki theory , jordan algebra , algebra representation , mathematical analysis , operations management , finance , economics
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.