# Pseudodifferential Analysis on Conformally Compact Spaces

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*Robert Lauter*

The \(0\)-calculus on a manifold with boundary is a micro-localization of the Lie algebra of vector fields that vanish at the boundary. It has been used by Mazzeo, Melrose to study the Laplacian of a conformally compact metric. We give a complete characterization of those \(0\)-pseudodifferential operators that are Fredholm between appropriate weighted Sobolev spaces, and describe \(C^{*}\)-algebras that are generated by \(0\)-pseudodifferential operators. An important step is understanding the so-called reduced normal operator, or, almost equivalently, the infinite dimensional irreducible representations of \(0\)-pseudodifferential operators. Since the \(0\)-calculus itself is not closed under holomorphic functional calculus, we construct submultiplicative Fréchet \(*\)-algebras that contain and share many properties with the \(0\)-calculus, and are stable under holomorphic functional calculus (\(\Psi^{*}\)-algebras in the sense of Gramsch). There are relations to elliptic boundary value problems.

#### Table of Contents

# Table of Contents

## Pseudodifferential Analysis on Conformally Compact Spaces

- Contents v6 free
- Introduction ix10 free
- Part 1. Predholm theory for 0-pseudodifferential operators 118 free
- Chapter 1. Review on basic objects of 0-geometry 320
- Chapter 2. The small 0-calculus and the 0-calculus with bounds 1128
- Chapter 3. The b-c-calculus on an interval 1936
- 3.1. The b-c-structure algebra 1936
- 3.2. The b-c-double space 1936
- 3.3. b-c-densities 2138
- 3.4. The b-c calculus with bounds 2138
- 3.5. Basic properties of the b-c-calculus 2239
- 3.6. Fredholm theory for the b-c-calculus 2441
- 3.7. Invariance of the b-c-calculus under the R[sub(+)]-action 2542
- 3.8. C*-algebras of b-c-operators 2643
- 3.9. General bundles 2744

- Chapter 4. The reduced normal operator 2946
- 4.1. Definition of the reduced normal operator 2946
- 4.2. Coordinate invariance of the reduced normal operator 3047
- 4.3. Scale invariance of the reduced normal operator 3148
- 4.4. Characterization of the reduced normal operator 3249
- 4.5. Basic properties of the reduced normal operator 4057
- 4.6. The case of 0-differential operators 4259
- 4.7. General bundles 4360

- Chapter 5. Weighted 0-Sobolev spaces 4562
- Chapter 6. Fredholm theory for 0-pseudodifferential operators 4966

- Part 2. Algebras of 0-pseudodifferential operators of order 0 5572
- Chapter 7. C*-algebras of 0-pseudodifferential operators 5774
- 7.1. Solvable C*-algebras 5774
- 7.2. The reduced normal operator on S*∂X 5774
- 7.3. Extension of the symbolic structure 5875
- 7.4. The C*-algebra generated by the reduced normal operator 5976
- 7.5. The C*-algebra B[sup((a))][sub(0)](X,[sup(0)]Ω[sup(1/2)]) 6279
- 7.6. The spectrum of the C*-algebra B[sup((a))][sub(0)](X,[sup(0)]Ω[sup(1/2)]) 6380

- Chapter 8. Ψ*-algebras of 0-pseudodifferential operators 6986

- Appendix A. Spaces of conormal functions 7390
- Bibliography 7996
- Notations 85102
- Index 89106