The $p$-Operator Approximation Property

TL;DR

This paper introduces the $p$-Operator Approximation Property ($p$-OAP) for operator spaces, providing multiple characterizations and exploring its transfer properties, with applications to approximation of Herz-Schur multipliers in noncommutative $C^*$-algebras.

Contribution

It defines the $p$-OAP in the noncommutative setting, establishes equivalent characterizations, and studies its transfer properties, extending the approximation theory in operator spaces.

Findings

Equivalent characterizations of $p$-OAP for operator spaces.

Transfer of $p$-OAP from duals to original spaces.

Approximation of operator $p$-compact Herz-Schur multipliers.

Abstract

We study a notion analogous to the $p$-Approximation Property ($p$-AP) for Banach spaces, within the noncommutative context of operator spaces. Referred to as the $p$-Operator Approximation Property ($p$-OAP), this concept is linked to the ideal of operator $p$-compact mappings. We present several equivalent characterizations based on the density of finite-rank mappings within specific spaces for different topologies, and also one in terms of a slice mapping property. Additionally, we investigate how this property transfers from the dual or bidual to the original space. As an application, the $p$-OAP for the reduced $C^*$-algebra of a discrete group implies that operator $p$-compact Herz-Schur multipliers can be approximated in $\mbox{cb}$-norm by finitely supported multipliers.