Embeddings of $L^p$-operator algebras

Abstract

We study embeddings of $L^p$-operator algebras arising from (twis\-ted) \'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 embeddability results both for spatial AF $L^p$-operator algebras and for tensor products of $L^p$-Cuntz algebras. For $p\not \in \{1,2\}$, a reduced $L^p$-groupoid algebra associated with a principal \'etale groupoid embeds into a spatial AF $L^p$-operator algebra if and only if the underlying groupoid is AF. In particular, and in contrast with classical results of Pimsner-Voiculescu, irrational $L^p$-noncommutative tori do not embed into spatial AF $L^p$-operator algebras for $p\neq 2$. Furthermore, if $p\neq 2$, 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.