K-theory of C*-algebras

From idempotents and vector bundles to exact sequences, Bott periodicity, trace determinants and crossed-product computations.

23 English lessons · 115 solved exercises · editable Markdown · CC0 1.0

This edition includes the checked free-source comparisons and the resulting proof expansions. Separate prerequisite gaps are identified in the preparation notes.

1. Idempotents, projections and their equivalences
2. Vector bundles and finitely generated projective modules
3. The Grothendieck group and \(K_0\) of a unital algebra
4. Nonunital algebras: unitization, relative classes and half-exactness
5. Matrix stability, stability and continuity of \(K_0\)
6. Invertibles, unitaries and \(K_1\)
7. The index map and the exact sequence at \(K_0\)
8. Suspension, higher K-groups and the long exact sequence
9. Toeplitz operators and the index theorem on the circle
10. Bott periodicity
11. The six-term exact sequence and the exponential map
12. Topological K-theory of spaces, pairs and vector bundles
13. Traces, states and the pairing with K-theory
14. Determinants of traces and the pairing with K_1
15. Smooth subalgebras and the density theorem
16. Inductive limits and the K-theory of AF and AT algebras
17. The mapping torus
18. The Pimsner–Voiculescu exact sequence
19. Traces on integer crossed products and their K-theory ranges
20. Irrational rotation algebras
21. Commutative and noncommutative tori
22. Cuntz algebras
23. Cuntz–Krieger algebras, minimal systems and projectionless algebras

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