Acknowledgements; Contents; 1 Introduction; 1.1 Algebraic Approach to Quantum Field Theory on Curved Spacetimes; 1.2 States with Good Physical Properties; 1.3 Bulk to Boundary Correspondence and Construction of Hadamard States; References; 2 General Geometric Setup; 2.1 Globally Hyperbolic Spacetimes; 2.2 Spacetimes with Distinguished Light-Like 3-Surfaces; 2.2.1 Asymptotically Flat Spacetimes; 2.2.2 A Compound Example: Schwarzschild Spacetime; 2.2.3 Asymptotic Symmetries: The Bondi-Metzner-Sachs Group; 2.2.4 Expanding Universes with Cosmological Horizon

2.2.5 The Cosmological-Horizon Symmetry GroupReferences; 3 Quantum Fields in Spacetimes with Null Surfaces; 3.1 Algebra of Observables in Globally Hyperbolic Spacetimes; 3.2 Observables on Null Infinity; 3.3 The Bulk-to-Boundary Algebra Injection; 3.3.1 Asymptotically Flat Spacetimes; 3.3.2 Cosmological Spacetimes; 3.3.3 The Case of Schwarzschild Spacetime; References; 4 Boundary-Induced Hadamard States; 4.1 Algebraic States and the Hadamard Condition; 4.1.1 Algebraic States, GNS Construction, and All That; 4.1.2 Hadamard States; 4.1.3 Natural Gaussian States for the Boundary Algebra

4.2 The Bulk-to-Boundary Correspondence for States4.2.1 Asymptotically Flat Spacetimes; 4.2.2 Cosmological Spacetimes; 4.2.3 The Unruh State on Schwarzschild Spacetime; 4.3 Some Further Applications; 4.3.1 Asymptotic Vacua and Scattering States; 4.3.2 Applications to Cosmology; 4.3.3 Hadamard States and Hawking Radiation; References; 5 Wick Polynomials and Extended Observables Algebras; 5.1 Extended Algebra of Fields; 5.2 Extension of the Algebra on the Boundary; 5.3 Interplay Between Algebras and States on mathscrD and on mathscrC; References