Embeddings of $L^p$-operator algebras

TL;DR

This paper investigates embeddings of $L^p$-operator algebras from groupoids, revealing rigidity phenomena and providing tools to determine when such algebras can embed into simpler or well-understood structures, especially for $p eq 2$.

Contribution

It introduces a detailed analysis of embeddings of $L^p$-groupoid algebras, linking them to groupoid morphisms, and establishes new rigidity results and non-embedding theorems for $p eq 2$.

Findings

Embeddings of $L^p$-groupoid algebras correspond to groupoid morphisms.

Irrational rotation $L^p$-operator algebras do not embed into spatial $L^p$-AF-algebras.

No unital contractive homomorphism exists from $ ext{O}_2^p imes_p ext{O}_2^p$ into $ ext{O}_2^p$ for $p eq 2$.

Abstract

We study embeddings of $L^p$-operator algebras arising from (twisted) \'etale groupoids, with particular emphasis on rigidity phenomena for $p\neq 2$. Our methods rely on a detailed analysis of core normalizers and their functorial behavior under algebra homomorphisms. Using the notion of actors between groupoids, we show that under natural hypotheses, embeddings between reduced $L^p$-groupoid algebras can be described entirely in terms of morphisms of the underlying groupoids. We further show that embeddings of $L^p$-groupoid algebras induce embeddings of the associated topological full groups. Our results provide new tools for studying embeddability questions in the $L^p$-setting, and are particularly helpful when ruling out the existence of embeddings. As applications, we obtain strong rigidity results for (spatial) $L^p$-AF-embeddability, showing that, for $p\neq 2$, an $L^p$-groupoid algebra embeds into a spatial $L^p$-AF-algebra if and only if the underlying groupoid is AF. In particular, irrational rotation $L^p$-operator algebras do not embed into spatial $L^p$-AF-algebras. We apply these results to tensor products of $L^p$-Cuntz algebras and prove that, for $p\neq 2$, there is no unital contractive homomorphism from $\mathcal{O}_2^p \otimes_p \mathcal{O}_2^p$ into $\mathcal{O}_2^p$, showing that there is no $L^p$-analog of Kirchberg's $\mathcal{O}_2$-embedding theorem.