# Arens Algebras and Matricial Spaces

## DOI:

https://doi.org/10.26577/JMMCS-2018-4-583## Keywords:

von Neumann algebra, finite trace, Arens “algebras”, Noncommutative Lp-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].

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*100*(4), 3–7. https://doi.org/10.26577/JMMCS-2018-4-583