Arens Algebras and Matricial Spaces

Abstract

Let M be a finite von Neumann algebra equipped with a finite faithful normal trace and let
Lp(M; ) be the corresponding noncommutative Lp space of -measurable operators associated
with the couple (M; ), 1 ≤ p < ∞. Let MN be the algebra of all complex N × N-matrices
equipped with the standard trace Tr. In this note we study the properties of Arens “algebras” over
finite 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-spaces
fails 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, in
contrast to L!(M; ): In particular, we provide an example of a finite von Neumann algebra, with
an associated trace, such that L!(; h) is not an algebra, for any choice of ∈ [0; 1].