Simplicity of UHF and Cuntz algebras on $L^p$~spaces

Abstract

We prove that, for $p\in [1,\infty ),$ and integers $d$ at least 2, the $L^p$ analog ${\mathcal{O}}_d^p$ of the Cuntz algebra ${\mathcal{O}}_d$ is a purely infinite simple amenable Banach algebra. The proof requires what we call the spatial $L^p$ UHF algebras, which are analogs of UHF algebras acting on $L^p$ spaces. As for the usual UHF C*-algebras, they have associated supernatural numbers. For fixed $p\in [1,\infty ),$ we prove that any spatial $L^p$ UHF algebra is simple and amenable, and that two such algebras are isomorphic if and only if they have the same supernatural number (equivalently, the same scaled ordered $K_0$-group). For distinct $p_1,p_2\in [1,\infty ),$ we prove that no spatial $L^{p_1}$ UHF algebra is isomorphic to any spatial $L^{p_2}$ UHF algebra.

AMSLaTeX; 36 pages. Changes from version 1: Lemma 1.4: Proof replaced by reference to literature. Theorem 3.7: Generalized to allow tensoring with a fixed algebra (for use in another paper). Misprints corrected

Cited by in corpus (5)

• Algebraically irreducible representations and structure space of the Banach algebra associated with a topological dynamical system
• Classification of $L^p$ AF algebras
• A purely infinite Cuntz-like Banach $\ast$-algebra with no purely infinite ultrapowers
• Structure and $K$-theory of ${\ell }^p$ uniform Roe algebras
• Representations of étale groupoids on $L^p$-spaces