Appalachian set theory : 2006-2012 / edited by James Cummings and Ernest Schimmerling.

Print version record.

Papers based on a series of workshops where prominent researchers present exciting developments in set theory to a broad audience.

Includes bibliographical references.

Cover; LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES; Title; Copyright; Contents; Contributors; Introduction; 1 An introduction to Pmax forcing; 1 Introduction; 2 Setup: iterations and the definition of Pmax; 3 First properties of Pmax; 4 Existence of Pmax conditions; 5 S2 maximality; 6 Discussion; References; 2 Countable Borel Equivalence Relations; First lecture; 1.1 Standard Borel spaces and Borel equivalence relations; 1.2 Borel reducibility; 1.3 Countable Borel equivalence relations; 1.4 Turing equivalence and the Martin conjectures; Second lecture.

2.1 The fundamental question in the theory of countable Borel equivalence relations2.2 Essentially free countable Borel equivalence relations; 2.3 Bernoulli actions, Popa superrigidity, and the proof of Theorem 2.11; 2.4 Free and non-essentially free countable Borel equivalence relations; Third lecture; 3.1 Ergodicity, strong mixing and Borel cocycles; 3.2 Popa's Cocycle Superrigidity Theorem and the proof of Theorem 2.16; 3.3 Torsion-free abelian groups of finite rank; 3.4 E0-ergodicity; 3.5 The non-universality of the isomorphism relation for torsion-free abelian groups of finite rank.

Fourth lecture4.1 Containment vs. Borel reducibility; 4.2 Unique ergodicity and ergodic components; 4.3 The proof of Theorem 4.5; 4.4 Profinite actions and Ioana superrigidity; Open problems; 5.1 Hyperfinite relations.; 5.2 Treeable relations.; 5.3 Universal relations.; References; 3 Set theory and operator algebras; Acknowledgments; 1 Introduction; 1.1 Nonseparable C*-algebras; 1.2 Ultrapowers; 1.3 Structure of corona algebras; 1.4 Classification and descriptive set theory; 2 Hilbert spaces and operators; 2.1 Normal operators and the spectral theorem; 2.2 The spectrum of an operator.

3 C*-algebrasTypes of operators in C*-algebras; 3.1 Some examples of C*-algebras; Full matrix algebras; The algebra of compact operators; The Calkin algebra; 3.2 Automatic continuity and the Gelfand transform; 3.3 Continuous functional calculus; 3.4 More examples of C*-algebras; Direct limits; UHF (uniformly hyperfinite) algebras; AF (approximately finite) algebras; Even more examples; 4 Positivity, states and the GNS construction; 4.1 Irreducible representations and pure states; 4.2 On the existence of states; 5 Projections in the Calkin algebra; Stone duality.

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