Model theory of C∗-algebras / Ilijas Farah, Bradd Hart, Martino Lupini, Leonel Robert, Aaron Tikuisis, Alessandro Vignati, Wilhelm Winter.

Format:

Book

Author/Creator:

Farah, Ilijas.

Contributor:

Hart, Bradd.

Lupini, Martino.

Series:

Memoirs of the American Mathematical Society ; no. 1324

Memoirs of the American Mathematical Society, 0065-9266 ; volume 271, number 1324

Language:

English

Subjects (All):

C*-algebras.

Model theory.

Physical Description:

1 online resource (142 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"A number of significant properties of 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 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 C- algebra, or the set of elements that are Cuntz-Pedersen equivalent to 0, depends on structural properties of the C-algebra in question. Regularity properties required in the Elliott programme for classification of nuclear C-algebras imply the definability of some of these sets. In fact any known pair of separable, nuclear, unital and simple 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 C-algebraic sense) 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 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 C-algebra"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Chapter 2. Continuous model theory

2.1. Preliminaries

2.2. Theories

2.3. Ultraproducts

2.3.1. Atomic and Elementary Diagrams

2.4. Elementary classes and preservation theorems

2.5. Elementary classes of \cstar-algebras

2.5.1. Abelian algebras

2.5.2. Non-abelian algebras

2.5.3. Real rank zero again

2.5.4. -subhomogeneous

2.5.5. Non- -subhomogeneous algebras

2.5.6. Tracial \cstar-algebras

2.5.7. \cstar-algebras with a character

2.6. Downward Löwenheim-Skolem

2.7. Tensorial absorption and elementary submodels

2.7.1. Strongly self-absorbing \cstar-algebras

2.7.2. Stable algebras

Chapter 3. Definability and ^{\eq}

3.1. Expanding the definition of formula: definable predicates and functions

3.1.1. Definable predicates

3.2. Expanding the definition of formula: definable sets

3.3. Expanding the language: imaginaries

Countable products

Definable sets

Quotients

^{\eq} and ^{\eq}

3.4. The use of continuous functional calculus

3.5. Definability of traces

3.5.1. Definability of Cuntz-Pedersen equivalence

3.6. Axiomatizability via definable sets

3.6.1. Projectionless and unital projectionless

3.6.2. Real rank zero revisited

3.6.3. Infinite \cstar-algebras

3.6.4. Finite and stably finite algebras

3.7. Invertible and non-invertible elements

3.8. Stable rank

3.9. Real rank

3.10. Tensor products

3.11. ₀( ) and ^{\eq}

3.12. ₁( ) and ^{\eq}

3.13. Co-elementarity

3.13.1. Abelian algebras

3.13.2. Infinite algebras

3.13.3. Algebras containing a unital copy of _{ }(\bbC)

3.13.4. Definability of sets of projections

3.13.5. Stable rank one

3.13.6. Real rank zero

3.13.7. Purely infinite simple \cstar-algebras

3.14. Some non-elementary classes of \cstar-algebras.

Chapter 4. Types

4.1. Types: the definition

4.1.1. Types as sets of conditions

4.2. Beth definability

4.3. Saturated models

4.4. MF algebras

4.5. Approximately divisible algebras

Chapter 5. Approximation properties

5.1. Nuclearity

5.2. Completely positive contractive order zero maps

5.3. Nuclear dimension

5.4. Decomposition rank

5.5. Quasidiagonal algebras

5.6. Approximation properties and definability

5.7. Approximation properties and uniform families of formulas

5.7.1. Uniform families of formulas

5.8. Nuclearity, nuclear dimension and decomposition rank: First proof

5.9. Nuclearity, nuclear dimension and decomposition rank: Second proof

5.10. Simple \cstar-algebras

5.11. Popa algebras

5.12. Simple tracially AF algebras

5.13. Quasidiagonality

5.14. An application: Preservation by quotients

5.15. An application: Perturbations

5.16. An application: Preservation by inductive limits

5.17. An application: Borel sets of \cstar-algebras

Chapter 6. Generic \cstar-algebras

6.1. Henkin forcing

6.2. Infinite forcing

6.3. Finite forcing

6.4. ∀∃-axiomatizability and existentially closed structures

6.5. Strongly self-absorbing algebras

6.6. Stably finite, quasidiagonal, and MF algebras

Chapter 7. \cstar-algebras not elementarily equivalent to nuclear \cstar-algebras

7.1. Exact algebras

7.2. Definability of traces: the uniform strong Dixmier property

7.3. Elementary submodels of von Neumann algebras

Chapter 8. The Cuntz semigroup

8.1. Cuntz subequivalence

8.2. Strict comparison of positive elements

8.3. The Toms-Winter conjecture

8.4. Radius of comparison

Appendix A. \cstar-algebras

Bibliography

Index

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references and index.

ISBN:

9781470466268

1470466260

OCLC:

1266905767