Contents Preface xi Part 1. Analysis and Geometry on Foliated Spaces Foreword to Part 1 3 Chapter 1. The C∗-Algebra of a Foliated Space 5 §1.1. Twisted Forms and Densities 6 §1.2. Functions on Non-Hausdorff Spaces 8 §1.3. The Graph of a Foliated Space 11 §1.4. The C∗-algebra of a Foliated Space 18 §1.5. The Basic Examples 27 §1.6. Quasi-invariant Currents 37 §1.7. Representations of the Foliation C∗-algebra 48 §1.8. Minimal Foliations and their C∗-algebras 54 Chapter 2. Harmonic Measures for Foliated Spaces 61 §2.1. Existence of Harmonic Measures 62 §2.2. The Diffusion Semigroup 68 §2.3. The Markov Process 80 §2.4. Characterizations of Harmonic Measures 86 §2.5. The Ergodic Theorem 96 §2.6. Ergodic Decomposition of Harmonic Measures 99 §2.7. Recurrence 112 vii
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