Embeddings of $L^p$-operator algebras
Eusebio Gardella, Jan Gundelach
TL;DR
This work develops an L^p-operator algebra framework for embeddings arising from twisted etale groupoids, introducing Weyl twists, core normalizers, and actor-based morphisms to translate algebra embeddings into underlying groupoid data. It proves a general rigidity principle: under mild hypotheses, unital contractive embeddings between reduced $L^p$-groupoid algebras are induced by groupoid morphisms, with consequences for the associated topological full groups and Cartan-type reconstructions. A central result is that, for $p\neq 2$, an $L^p$-groupoid algebra embeds into a spatial $L^p$-AF-algebra only if the groupoid is AF, implying strong obstructions for irrational rotation algebras and ruling out $O_2$-type embeddings in the $L^p$ setting. The paper further links embeddability to topological full groups, analyzes the structure of invertible isometries, and applies the theory to tensor products of $L^p$-Cuntz algebras, establishing a robust rigidity theory in the non-Hilbertian regime and highlighting fundamental differences from the C*-algebra world.
Abstract
We study embeddings of $L^p$-operator algebras arising from (twisted) étale 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.