Classification of L^p AF algebras

2017/07/28 by N. Christopher Phillips, Phillips, N. Christopher, Maria Grazia Viola +1
Mathematics · #46L35 #47L10 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Banach algebra #Banach space #Countable set #Discrete mathematics #Division algebra #FOS: Mathematics #Filtered algebra #Group (periodic table) #Mathematics #Normed algebra #Operator Algebras (math.OA) #Operator algebra #Physics #Pure mathematics #math.OA #msc:46L35 #msc:47L10

paper · pdf · doi:10.48550/arxiv.1707.09257

42 pages, added one comment to previous version of article

openalex publication_date 2017/07/28 · arxiv created 2017/10/06 · arxiv updated 2017/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define spatial L^p AF algebras for p ∈ [1, ∞) ∖ \ 2 \, and prove the following analog of the Elliott AF algebra classification theorem. If A and B are spatial L^p AF algebras, then the following are equivalent: 1) A and B have isomorphic scaled preordered K₀-groups. 2) A ≅ B as rings. 3) A ≅ B (not necessarily isometrically) as Banach algebras. 4) A is isometrically isomorphic to B as Banach algebras. 5) A is completely isometrically isomorphic to B as matrix normed Banach algebra. As background, we develop the theory of matrix normed L^p operator algebras, and show that there is a unique way to make a spatial L^p AF algebra into a matrix normed L^p operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered K₀-group of a spatial L^p AF algebra.

Citations

• Crossed products of L^p operator algebras and the K-theory of Cuntz algebras on L^p spaces

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