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Model theory of $\mathrm {C}^*$-algebras
About this Title
Ilijas Farah, Bradd Hart, Martino Lupini, L. Robert, Aaron Tikuisis, Alessandro Vignati and Wilhelm Winter
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 271, Number 1324
ISBNs: 978-1-4704-4757-1 (print); 978-1-4704-6626-8 (online)
DOI: https://doi.org/10.1090/memo/1324
Published electronically: June 30, 2021
Keywords: Model theory,
continuous logic,
nuclear $\mathrm {C}^*$-algebras,
model theoretic forcing
Table of Contents
Chapters
- 1. Introduction
- 2. Continuous model theory
- 3. Definability and $A^{eq}$
- 4. Types
- 5. Approximation properties
- 6. Generic $\mathrm {C}^*$-algebras
- 7. $\mathrm {C}^*$-algebras not elementarily equivalent to nuclear $\mathrm {C}^*$-algebras
- 8. The Cuntz semigroup
- A. $\mathrm {C}^*$-algebras
Abstract
A number of significant properties of $\mathrm {C}^*$-algebras can be expressed in continuous logic, or at least in terms of definable (in a model-theoretic sense) sets. Certain sets, such as the set of projections or the unitary group, are uniformly definable across all $\mathrm {C}^*$-algebras. On the other hand, the definability of some other sets, such as the connected component of the identity in the unitary group of a unital $\mathrm {C}^*$-algebra, or the set of elements that are Cuntz–Pedersen equivalent to $0$, depends on structural properties of the $\mathrm {C}^*$-algebra in question. Regularity properties required in the Elliott programme for classification of nuclear $\mathrm {C}^*$-algebras imply the definability of some of these sets. In fact any known pair of separable, nuclear, unital and simple $\mathrm {C}^*$-algebras with the same Elliott invariant can be distinguished by their first-order theory. Although parts of the Elliott invariant of a classifiable (in the technical $\mathrm {C}^*$-algebraic sense) $\mathrm {C}^*$-algebra can be reconstructed from its model-theoretic imaginaries, the information provided by the theory is largely complementary to the information provided by the Elliott invariant. We prove that all standard invariants employed to verify non-isomorphism of pairs of $\mathrm {C}^*$-algebras indistinguishable by their K-theoretic invariants (the divisibility properties of the Cuntz semigroup, the radius of comparison, and the existence of finite or infinite projections) are invariants of the theory of a $\mathrm {C}^*$-algebra.
Many of our results are stated and proved for arbitrary metric structures. We present a self-contained treatment of the imaginaries (most importantly, definable sets and quotients) and a self-contained description of the Henkin construction of generic $\mathrm {C}^*$-algebras and other metric structures.
Our results readily provide model-theoretic reformulations of a number of outstanding questions at the heart of the structure and classification theory of nuclear $\mathrm {C}^*$-algebras. The existence of counterexamples to the Toms–Winter conjecture and some other outstanding conjectures can be reformulated in terms of the existence of a model of a certain theory that omits a certain sequence of types. Thus the existence of a counterexample is equivalent to the assertion that the set of counterexamples is generic. Finding interesting examples of $\mathrm {C}^*$-algebras is in some cases reduced to finding interesting examples of theories of $\mathrm {C}^*$-algebras. This follows from one of our main technical results, a proof that some non-elementary (in model-theoretic sense) classes of $\mathrm {C}^*$-algebras are ‘definable by uniform families of formulas.’ This was known for UHF and AF algebras, and we extend the result to $\mathrm {C}^*$-algebras that are nuclear, have nuclear dimension $\leq n$, decomposition rank $\leq n$, are simple, are quasidiagonal, Popa algebras, and are tracially AF. We show that some theories of $\mathrm {C}^*$-algebras do not have nuclear models. As an application of the model-theoretic vantage point, we give a proof that various properties of $\mathrm {C}^*$-algebras are preserved by small perturbations in the Kadison–Kastler distance.
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