Arens Algebras and Matricial Spaces
AbstractLet 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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