# Arens Algebras and Matricial Spaces

### 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].### References

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states,"Positivity. 14 (2010): no.1, 105-121.

[2] Arens R. "The space Lw and convex topological rings,"Bull. Amer. Math. Soc. 52 (1946), 931-935.

[3] Connes A. "On the spatial theory of von Neumann algebras,"J. Funct. Anal. 35 (1980): no.2, 153-164.

[4] Fack T. and Kosaki H. "Generalized s-numbers of -measurable operators,"Pacific J. Math. 123 (1986): no.2, 269-300.

[5] Haagerup U. "Lp-spaces associated with an arbitrary von Neumann algebra,"Proc. Colloq. Internat. CNRS. 274 (1979),

175-184.

[6] Hilsum M. "Les espaces Lp d’une algebre de von Neumann definies par la deriv ee spatiale,"J. Funct. Anal. 40 (1981):

no.2, 151-169.

[7] Kunze W. "Noncommutative Orlicz spaces and generalized Arens algebras,"Math. Nachr. 147 (1990), 123-138.

[8] S. Lord, F. Sukochev, and D. Zanin: Singular traces. Theory and applications (De Gruyter, Berlin, 2013).

[9] M. Takesaki: Theory of operator algebras. II. (Springer-Verlag, Berlin, 2003).

[10] M. Terp: Lp spaces associated with von neumann algebras (Notes, Math. Institute, Copenhagen Univ., 1981).

[11] Trunov N. V. "Integration in von Neumann algebras and regular weights,"In Constructive theory of functions and

functional analysis. Kazan. Gos. Univ., Kazan. (1981): no.3, 73-87.

[12] Trunov N. V. "Lp spaces associated with a weight on a semifinite von Neumann algebra,"In Constructive theory of

functions and functional analysis. Kazan. Gos. Univ., Kazan. (1981): no.3, 88-93.

states,"Positivity. 14 (2010): no.1, 105-121.

[2] Arens R. "The space Lw and convex topological rings,"Bull. Amer. Math. Soc. 52 (1946), 931-935.

[3] Connes A. "On the spatial theory of von Neumann algebras,"J. Funct. Anal. 35 (1980): no.2, 153-164.

[4] Fack T. and Kosaki H. "Generalized s-numbers of -measurable operators,"Pacific J. Math. 123 (1986): no.2, 269-300.

[5] Haagerup U. "Lp-spaces associated with an arbitrary von Neumann algebra,"Proc. Colloq. Internat. CNRS. 274 (1979),

175-184.

[6] Hilsum M. "Les espaces Lp d’une algebre de von Neumann definies par la deriv ee spatiale,"J. Funct. Anal. 40 (1981):

no.2, 151-169.

[7] Kunze W. "Noncommutative Orlicz spaces and generalized Arens algebras,"Math. Nachr. 147 (1990), 123-138.

[8] S. Lord, F. Sukochev, and D. Zanin: Singular traces. Theory and applications (De Gruyter, Berlin, 2013).

[9] M. Takesaki: Theory of operator algebras. II. (Springer-Verlag, Berlin, 2003).

[10] M. Terp: Lp spaces associated with von neumann algebras (Notes, Math. Institute, Copenhagen Univ., 1981).

[11] Trunov N. V. "Integration in von Neumann algebras and regular weights,"In Constructive theory of functions and

functional analysis. Kazan. Gos. Univ., Kazan. (1981): no.3, 73-87.

[12] Trunov N. V. "Lp spaces associated with a weight on a semifinite von Neumann algebra,"In Constructive theory of

functions and functional analysis. Kazan. Gos. Univ., Kazan. (1981): no.3, 88-93.

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 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/583>. Date accessed: 20 jan. 2021. doi: https://doi.org/10.26577/JMMCS-2018-4-583.
Section

Mathematics

Keywords
von Neumann algebra, finite trace, Arens “algebras”, Noncommutative Lp-spaces