Classification of $L^p$ AF algebras

TL;DR

This paper extends the classification of AF algebras to spatial $L^p$ AF algebras for $p$ in [1, ∞) excluding 2, establishing equivalences based on $K_0$-groups and various algebraic isomorphisms.

Contribution

It introduces spatial $L^p$ AF algebras, develops their matrix normed $L^p$ operator algebra theory, and proves a classification theorem analogous to Elliott's for these algebras.

Findings

Spatial $L^p$ AF algebras are classified by their scaled preordered $K_0$-groups.

Unique matrix normed $L^p$ operator algebra structure on spatial $L^p$ AF algebras.

Any countable scaled Riesz group can be realized as a $K_0$-group of a spatial $L^p$ AF algebra.

Abstract

We define spatial $L^p$ AF algebras for $p \in [1, \infty) \setminus \{ 2 \}$, 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 $K_0$-groups. 2) $A \cong B$ as rings. 3) $A \cong B$ (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 $K_0$-group of a spatial $L^p$ AF algebra.