Open AccessRigid $\mathcal{OL}_p$structures of non-commutative $L_p$-spaces associated with hyperfinite von Neumann algebras
Author(s) -
Marius Junge,
Zhong Jin Ruan,
Quanhua Xu
Publication year - 2005
Publication title -
mathematica scandinavica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.553
H-Index - 30
eISSN - 1903-1807
pISSN - 0025-5521
DOI - 10.7146/math.scand.a-14945
Subject(s) - mathematics , commutative property , space (punctuation) , von neumann algebra , von neumann architecture , pure mathematics , matrix (chemical analysis) , operator (biology) , operator algebra , combinatorics , philosophy , linguistics , biochemistry , chemistry , repressor , transcription factor , gene , materials science , composite material
This paper is devoted to the study of rigid local operator space structures on non-commutative $L_p$-spaces. We show that for $1\le p \neq 2 < \infty$, a non-commutative $L_p$-space $L_p(\mathcal M)$ is a rigid $\mathcal{OL}_p$ space (equivalently, a rigid $\mathcal{COL}_p$ space) if and only if it is a matrix orderly rigid $\mathcal{OL}_p$ space (equivalently, a matrix orderly rigid $\mathcal{COL}_p$ space). We also show that $L_p(\mathcal M)$ has these local properties if and only if the associated von Neumann algebra $\mathcal M$ is hyperfinite. Therefore, these local operator space properties on non-commutative $L_p$-spaces characterize hyperfinite von Neumann algebras.