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## The Heisenberg Lie algebra and its representations

### Link between Toeplitz operators on the Fock spaces and Weyl operators

#### Link with Toeplitz operators and an example of quantization

**A spectral triple from the Berezin–Toeplitz quantization**

**Table of contents :**

Introduction

**1 Toeplitz operators **

1.1 Bergman, Hardy and Fock spaces, Heisenberg algebra

1.1.1 Bergman and Hardy spaces

1.1.2 Fock space

1.1.3 The Heisenberg Lie algebra and its representations

1.2 Toeplitz operators

1.2.1 Classical Toeplitz operators

1.2.2 Generalized Toeplitz operators

1.3 Relations between Toeplitz operators

1.3.1 Link between Toeplitz on Bergman and GTOs

1.3.2 Toeplitz operators over Bn as elements of hn

1.3.3 Link between Toeplitz operators on the Fock spaces and Weyl operators

1.3.4 A diagram as a summary

1.4 Examples

1.4.1 GTO-like operators on the Bergman space

1.4.2 Unitary operators

**2 Quantization **

2.1 Geometric quantization

2.1.1 The different steps

2.1.2 Link with Toeplitz operators and an example of quantization

2.1.3 Approximating the classical observables

2.2 Deformation quantization

2.2.1 Berezin quantization

2.2.2 Berezin–Toeplitz quantization

**3 Noncommutative geometry **

3.1 Duality between topology and algebra

3.2 The Dirac operator and its properties

3.2.1 Construction

3.2.2 Hearing the shape of the manifold

3.3 Main tools in noncommutative geometry

3.3.1 Spectral triples

3.3.2 The spectral action

**4 Applications in noncommutative geometry **

4.1 Hardy space and spectral triples

4.1.1 A generic result

4.1.2 Examples of operators D

4.2 Bergman spaces and spectral triples

4.2.1 Over a general strictly pseudoconvex domain

4.2.2 Examples of operators D

4.2.3 The case of the unit ball

4.3 Remarks on Dixmier traces

4.4 A spectral triple for the Fock space over C

4.5 Links with quantization

4.5.1 A spectral triple from the Berezin–Toeplitz quantization

4.5.2 (Des)Integration of spectral triples

Conclusion and perspectives

**A Geometric framework **

A.1 Symplectic manifolds

A.2 Contact manifolds

A.3 Complex manifolds

A.4 Kähler manifolds

A.5 Pseudoconvex manifolds

A.6 Diagram

**B Biholomorphically invariant defining functions and logarithmic divergences **

**C Pseudodifferential operators **

Notations and symbols

**Bibliography**