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

Zhen Wang

Vol. 342 (2026), No. 2, 427–440

DOI: 10.2140/pjm.2026.342.427

Abstract

Let $p\in (1,\infty )$. 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 $F_{\lambda }^p(G)$, where $G$ is a discrete group, is $p$-nuclear and the $p$-pseudomeasure algebra $P{M}_p(G)$ is $p$-semidiscrete if $G$ is amenable. In this paper, we show that the following are equivalent: (i) $G$ is amenable; (ii) the reduced group $L^p$-operator algebra $F_{\lambda }^p(G)$ is $p$-nuclear; (iii) the $p$-pseudomeasure algebra $P{M}_p(G)$ is $p$-semidiscrete. This solves an open problem raised by N. C. Phillips concerning the $p$-nuclearity for reduced group $L^p$-operator algebras.

Keywords

reduced group $L^p$-operator algebra, $p$-pseudomeasure algebra, nuclearity, amenability

Mathematical Subject Classification

Primary: 47L10, 47L25

Secondary: 43A07, 43A15

Milestones

Received: 23 November 2025

Revised: 25 February 2026

Accepted: 26 February 2026

Published: 23 April 2026