$p$-nuclearity of reduced group $L^p$-operator algebras

TL;DR

This paper proves that the reduced group $L^p$-operator algebra is $p$-nuclear if and only if the underlying group is amenable, resolving an open problem in the field.

Contribution

It establishes the equivalence between $p$-nuclearity and amenability for reduced group $L^p$-operator algebras, completing the characterization.

Findings

$p$-nuclearity implies amenability of the group

The converse holds: amenability implies $p$-nuclearity

Answers an open problem posed by N. C. Phillips

Abstract

Let $p\in (1,\infty)$ and let $G$ be a discrete group. G. An, J.-J. Lee and Z.-J. Ruan introduced $p$-nuclearity for $L^p$-operator algebras. They proved that the reduced group $L^p$-operator algebra $F^p_\lambda(G)$ is $p$-nuclear if $G$ is amenable. In this paper, we show that the converse is true. This answers an open problem concerning the $p$-nuclearity for reduced group $L^p$-operator algebras of N. C. Phillips.