Isomorphism, nonisomorphism, and amenability of L^p UHF algebras

TL;DR

This paper explores a broader class of L^p UHF algebras constructed via diagonal similarities, revealing their nonisomorphism, amenability, and the conditions under which they are spatial, with implications for classification and structure.

Contribution

It introduces and characterizes a larger class of L^p UHF algebras, demonstrating their nonisomorphism and linking amenability and tensor flip properties to spatiality.

Findings

Uncountably many nonisomorphic L^p UHF algebras of tensor product type.

Spatial L^p UHF algebras are characterized by amenability and approximately inner tensor flip.

Conditions for isomorphism to spatial algebras are equivalent to amenability and a numerical criterion.

Abstract

In a previous paper, we introduced L^p UHF algebras for p in [1, \infty). We concentrated on the spatial L^p UHF algebras, which are classified up to isometric isomorphism by p and the scaled ordered K_0-group. In this paper, we concentrate on a larger class, the L^p UHF algebras of tensor product type constructed using diagonal similarities. Such an algebra is still simple and has the same K-theory as the corresponding spatial L^p UHF algebra. For each choice of p and the K-theory, we provide uncountably many nonisomorphic such algebras. We further characterize the spatial algebras among them. In particular, if A is one of these algebras, then A is isomorphic (not necessarily isometrically) to a spatial L^p UHF algebra if and only if A is amenable as a Banach algebra, also if and only if A has approximately inner tensor flip. These conditions are also equivalent to a natural numerical condition defined in terms of the ingredients used to construct the algebra.