Classification of O∞ -stable C*-algebras / by James Gabe.

Format:

Book

Author/Creator:

Gabe, James, author.

Series:

Memoirs of the American Mathematical Society ; 0065-9266 v. 1461.

Memoirs of the American Mathematical Society, 0065-9266 ; no. 1461

Language:

English

Subjects (All):

Operator algebras.

Physical Description:

v, 115 pages : illustrations ; 26 cm.

Other Title:

Classification of O infinity-stable C-star algebras

Place of Publication:

Providence, RI : American Mathematical Society, [2024]

Summary:

I present a proof of Kirchberg's classification theorem: two separable, nuclear, O∞ -stable C*-algebras are stably isomorphic if and only if they are ideal-related KK-equivalent. In particular, this provides a more elementary proof of the Kirchberg-Phillips theorem which is isolated in the paper to increase readability of this important special case.

Contents:

Chapter 1. Introduction and main results; Classification of C*-algebras through classification of maps ; The main results - the simple case ; The main results 0 the general case ; K-theoretic classification ; Notation ; Acknowledgements

Chapter 2. Equivalence of *-homomorphisms

Chapter 3. Approximate domination and nuclearity

Chapter 4. O2-stable and O∞-stable *-homomorphisms

Chapter 5. Absorbing representations ; 5.1. Cuntz sums and infinite repeats ; 5.2. General absorption ; 5.3. Nuclear absorption

Chapter 6. Asymptotic intertwining

Chapter 7. A unitary path and some key lemmas

Chapter 8. The Kirchberg-Phillips Theorem ; 8.1. KK-preliminaries ; 8.2. Proofs of Theorems A, B, and C ; 8.3. Kirchberg's Phillips' existence and uniqeueness results ; 8.4. Approximate equivalence

Chapter 9. Strongly O∞stable *-homomorphisms

Chapter 10. Ideals and actions of topological spaces ; 10.1. Ideal lattices ; 10.2. Actions of topological spaces on C*-algebras ; 10.3. X-fullness ; 10.4. Tensor products ; 10.5. X-nuclear maps

Chapter 11. Absorbing representations revisited

Chapter 12. Ideal-related KK-theory ; 12.1. The basics ; 12.2. The Kasparov product ; 12.3. The Cuntz pair picture ; 12.4. The Fredholm picture

Chapter 13. A stable uniqueness theorem

Chapter 14. An ideal-related O2-embedding theorem

Chapter 15. The main theorems ; 15.1. Existence of 0 ; 15.2. Proving Theorems E, F, and G ; 15.3. Approximate equivalence

Bibliography.

Notes:

"January 2024, volume 293, number 1461 (fifth of 7 numbers)."

Includes bibliographical references (pages 111-115).

ISBN:

1470467933

9781470467937

OCLC:

1420316669