Rigidity results for $L^p$-operator algebras and applications

Abstract

For $p\in [1,\infty )$, we show that every unital $L^p$-operator algebra contains a unique maximal $C^{\ast }$-subalgebra, which is always abelian if $p\ne 2$. Using this, we canonically associate to every unital $L^p$-operator algebra $A$ an étale groupoid ${\mathcal{G}}_A$, which in many cases of interest is a complete invariant for $A$. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case $p=2$; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the $L^p$-analog ${\mathcal{O}}_2^p$ of the Cuntz algebra, there is no isometric isomorphism between ${\mathcal{O}}_2^p$ and ${\mathcal{O}}_2^p{\otimes }^p{\mathcal{O}}_2^p$, when $p\ne 2$. In particular, we deduce that there is no $L^p$-version of Kirchberg's absorption theorem, and that there is no $K$-theoretic classification of purely infinite simple amenable $L^p$-operator algebras for $p\ne 2$. Our methods also allow us to recover a folklore fact in the case of C*-algebras ($p=2$), namely that no isomorphism ${\mathcal{O}}_2^p\cong {\mathcal{O}}_2^p\otimes {\mathcal{O}}_2^p$ preserves the canonical Cartan subalgebras.

v3: minor changes throughout. 35 pages. To appear in Advances in Math

References in corpus (12)

• Group algebras acting on $L^p$-spaces
• Representations of $p$-convolution algebras on $L^q$-spaces
• Tensor products of Leavitt path algebras
• Rigidity of ${\ell }^p$ Roe-type algebras
• L^p operator algebras with approximate identities I
• Cantor systems and quasi-isometry of groups
• Banach algebras generated by an invertible isometry of an $L^p$-space
• On convoluters on $L^p$-spaces
• Nonclassifiability of UHF $L^p$-operator algebras
• On the uniform Roe algebra as a Banach algebra and embeddings of ${\ell }_p$ uniform Roe algebras
• A purely infinite Cuntz-like Banach $\ast$-algebra with no purely infinite ultrapowers
• Directly finite algebras of pseudofunctions on locally compact groups

Cited by in corpus (6)

• Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
• On the topological ranks of Banach ${}^{\ast }$-algebras associated with groups of subexponential growth
• $L^p$-spectral triples and $p$-quantum compact metric spaces
• Twisted crossed products of Banach algebras
• C*-like modules and matrix $p$-operator norms
• $L^p$-modules and $L^p$-correspondences