Arens Algebras and Matricial Spaces

  • Denis Potapov School of Mathematics and Statistics, University of New South Wales, Kensington
  • Fedor Sukochev School of Mathematics and Statistics, University of New South Wales, Kensington

Abstract

Let M be a finite von Neumann algebra equipped with a finite faithful normal trace and letLp(M; ) be the corresponding noncommutative Lp space of -measurable operators associatedwith the couple (M; ), 1 ≤ p < ∞. Let MN be the algebra of all complex N × N-matricesequipped with the standard trace Tr. In this note we study the properties of Arens “algebras” overfinite dimensional matrix spaces, given by Trunov’s construction for noncommutative Lp-spaces.In this work we show that the Arens “algebras” built upon Trunov’s noncommutative Lp-spacesfails to form an algebra in general. We also show that the Arens space L!(; h), with 0 ≤ ≤ 1,fails to form an algebra in general, even in the setting of finite algebras associated to a trace, incontrast to L!(M; ): In particular, we provide an example of a finite von Neumann algebra, withan associated trace, such that L!(; h) is not an algebra, for any choice of ∈ [0; 1].

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Published
2019-01-22
How to Cite
POTAPOV, Denis; SUKOCHEV, Fedor. Arens Algebras and Matricial Spaces. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 100, n. 4, p. 3-7, jan. 2019. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/583>. Date accessed: 21 apr. 2019.
Keywords von Neumann algebra, finite trace, Arens “algebras”, Noncommutative Lp-spaces