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.2000 Mathematics Subject Classification. Primary 47L10, 03E15; Secondary 47L30.

Rigidity results for $L^p$-operator algebras and applications

“…The work of Phillips motivated other authors to study L p -analogs of other wellstudied families of C * -algebras. These classes include group algebras [30, 16,19]; groupoid algebras [14]; crossed products by topological systems [30]; AF-algebras [31,13]; and graph algebras [4]. In these works, an L p -operator algebra is obtained from combinatorial or dynamical data, and properties of the underlying data (such as hereditary saturation of a graph, or minimality of an action) are related to properties of the algebra (such as simplicity).…”

A modern look at algebras of operators onLp-spaces

“…The work of Phillips motivated other authors to study L p -analogs of well-studied families of C * -algebras. These classes include group algebras [26, 16,20]; groupoid algebras [14]; crossed products by topological systems [26]; AF-algebras [27,13]; and graph algebras [4]. In these works, an L p -operator algebra is obtained from combinatorial or dynamical data, and properties of the underlying data are related to properties of the algebra.…”

“…13. F p λ (Z 2 ) is the Banach subalgebra of B(ℓ p ({0, 1})) generated by the can be identified with C 2 , but its norm is not the maximum norm.…”