## Analysis over $$C^*$$-algebras and the oscillatory representation.(English)Zbl 1311.81159

The paper is of expository character and contains very few proofs (but provides references where they can be found). The paper covers the definition of symplectic structures and the metaplectic group $$\text{Mp}(V,\omega)$$ for a real symplectic vector space $$(V,\omega)$$ of dimension $$2n$$, Hodge theory of elliptic complexes and the definition of the Dirac operator using a symplectic connection on a symplectic manifold $$(M,\omega)$$. Finally, $$C^*$$-algebras and Hilbert $$C^*$$-modules are introduced.

### MSC:

 81S10 Geometry and quantization, symplectic methods 53D05 Symplectic manifolds (general theory) 14F40 de Rham cohomology and algebraic geometry 58J10 Differential complexes

### Keywords:

$$C^*$$-algebra; Hilbert $$C^*$$-modules; metaplectic group