Nonclassifiability of UHF 𝐿^{𝑝}-operator algebras

Gardella, Eusebio; Lupini, Martino

doi:10.1090/proc/12859

Nonclassifiability of UHF $L^p$-operator algebras
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by Eusebio Gardella and Martino Lupini PDF

Proc. Amer. Math. Soc. 144 (2016), 2081-2091 Request permission

Abstract:

For $p\in [1,\infty )$, 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.

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