Nonclassifiability of UHF $L^p$-operator algebras

TL;DR

This paper demonstrates that simple, separable UHF $L^p$-operator algebras are too complex to be classified using countable invariants, even in special cases, highlighting their high structural complexity.

Contribution

It proves nonclassifiability of UHF $L^p$-operator algebras by countable structures, extending to nonselfadjoint cases and answering a question by Phillips.

Findings

UHF $L^p$-operator algebras are not classifiable by countable structures.

Nonselfadjoint UHF operator algebras are also nonclassifiable.

Results rely on Borel complexity theory and Hjorth's turbulence.

Abstract

We prove that simple, separable, monotracial UHF $L^{p}$-operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF $L^{p}$-operator algebras of tensor product type obtained from a diagonal system of similarities. For $p=2$, it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of N. Christopher Phillips, rely on Borel complexity theory, and particularly Hjorth's theory of turbulence.