Approximation Properties for Non-commutative L_p-Spaces Associated with Discrete Groups

2002/06/06 by Marius Junge, M. Junge, Junge, M. +3 · 1 citation
Mathematics · #22D05 #46L07 #46L51 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:22D05 #msc:46L07 #msc:46L51

paper · pdf · doi:10.48550/arxiv.math/0206060

arxiv created 2002/06/06 · openalex publication_date 2002/06/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let 1 < p < ∞. It is shown that if G is a discrete group with the approximation property introduced by Haagerup and Kraus, then the non-commutative L_p(VN(G)) space has the operator space approximation property. If, in addition, the group von Neumann algebra VN(G) has the QWEP, i.e. is a quotient of a C^*-algebra with Lance's weak expectation property, then L_p(VN(G)) actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on L_p(VN(G)). Finally, we show that if G is a countable discrete group having the approximation property and VN(G) has the QWEP, then L_p(VN(G)) has a very nice local structure, i.e. it is a \mathcal C\OL_p space and has a completely bounded Schauder basis.

Citations

• Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups
• Arithmetic Subgroups of Algebraic Groups

Cited by

• The impact of Schur multipliers in harmonic analysis and operator algebras

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