Non commutative Lp spaces without the completely bounded approximation property

TL;DR

This paper constructs examples of non-commutative Lp spaces associated with lattices in special linear groups over local fields that lack the completely bounded approximation property, revealing new insights into operator algebra properties.

Contribution

It provides the first known examples of non-commutative Lp spaces without the completely bounded approximation property for p not equal to 2, especially in the context of lattices in SL_r over local fields.

Findings

Examples of non-commutative Lp spaces without the CBAP for p ≠ 2.

Lattices in SL_r(F) and SL_r(ℝ) do not have the Haagerup-Kraus approximation property.

These examples include exact C*-algebras lacking the operator space approximation property.

Abstract

For any 1\leq p \leq \infty different from 2, we give examples of non-commutative Lp spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3 these examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for r large enough depending on p. We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have the Approximation Property of Haagerup and Kraus. This provides examples of exact C^*-algebras without the operator space approximation property.