Nonseparable UHF algebras II: Classification

TL;DR

This paper constructs a vast family of nonisomorphic simple AF algebras and hyperfinite II$_1$ factors with specified density characters, showing these are maximal in number despite sharing invariants.

Contribution

It demonstrates the existence of $2^old$ nonisomorphic simple AF algebras and hyperfinite II$_1$ factors for each uncountable cardinal, with identical invariants.

Findings

Maximal number of nonisomorphic algebras at each density character

All constructed algebras share the same Elliott invariant and Cuntz semigroup as the CAR algebra

Establishes the diversity of operator algebras beyond invariant classification

Abstract

For every uncountable cardinal $\kappa$ there are $2^\kappa$ nonisomorphic simple AF algebras of density character $\kappa$ and $2^\kappa$ nonisomorphic hyperfinite II$_1$ factors of density character $\kappa$. These estimates are maximal possible. All C*-algebras that we construct have the same Elliott invariant and Cuntz semigroup as the CAR algebra.