Nonclassifiability of UHF $L^p$-operator algebras

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.

10 pages

Cited by in corpus (3)

• Rigidity results for $L^p$-operator algebras and applications
• $L^p$-spectral triples and $p$-quantum compact metric spaces
• C*-like modules and matrix $p$-operator norms