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

2013/08/31 by N. Christopher Phillips, Phillips, N. Christopher
Mathematics · #46H20 (Primary) 46H05 #47L10 (Secondary) #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Combinatorics #Computer science #Discrete mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Mathematics #Pure mathematics #Simple (philosophy) #Ultra high frequency #math.FA #msc:46H05 #msc:46H20 #msc:47L10

paper · pdf · doi:10.48550/arxiv.1309.0115

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

openalex publication_date 2013/08/31 · arxiv created 2013/09/14 · arxiv updated 2013/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, for p ∈ [1, ∞), and integers d at least 2, the L^p analog O_d^p of the Cuntz algebra 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 ∈ [1, ∞), 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 K₀-group). For distinct p₁, p₂ ∈ [1, ∞), we prove that no spatial L^p₁ UHF algebra is isomorphic to any spatial L^p₂ UHF algebra.

Citations

• A class ofC *-algebras and topological Markov chains

Related

• Isomorphism, nonisomorphism, and amenability of L^p UHF algebras
• Analogs of Cuntz algebras on L^p spaces
• Classification of L^p AF algebras
• Nonseparable UHF algebras I: Dixmier's problem
• Nonseparable UHF algebras II: Classification