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Classification of spatial $L^{p}$ $L^{p}$ AF algebras

N. Christopher Phillips

Department of Mathematics, University of Oregon, Eugene OR 97403-1222, USA

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and

Maria Grazia Viola

Lakehead University Orillia, Orillia ON L3V 0B9, Canada

Fields Institute, 222 College Street, Toronto, ON M5T 3J1, Canada

E-mail Address: mviola@lakeheadu.ca

Corresponding author.

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https://doi.org/10.1142/S0129167X20500883Cited by:0 (Source: Crossref)

Abstract

We define spatial $L^{p}$ $L^{p}$ AF algebras for $p \in [1, \infty) ∖ {2}$ , 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:

$A$ and $B$ have isomorphic scaled preordered $K_{0}$ -groups.
$A ≅ B$ as rings.
$A ≅ B$ (not necessarily isometrically) as Banach algebras.
$A$ is isometrically isomorphic to $B$ as Banach algebras.
$A$ is completely isometrically isomorphic to $B$ as matricial $L^{p}$ operator algebras.

As background, we develop the theory of matricial $L^{p}$ operator algebras, and show that there is a unique way to make a spatial $L^{p}$ AF algebra into a matricial $L^{p}$ operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered $K_{0}$ -group of a spatial $L^{p}$ AF algebra.

Keywords:

AMSC: Primary: 47L10, Secondary: 46L35

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