Applications of Connes' Geodesic Flow to Trace Formulas in Noncommutative Geometry

✍ Scribed by François Golse; Eric Leichtnam

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 428 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3305

✦ Synopsis

The ``trace formula'' of Chazarain, Duistermaat, and Guillemin expresses that the singularities of the distribution trace of the wave group on a compact Riemannian manifold X is included in the set of periods of the geodesic flow restricted to S*X. Most of the objects involved in this trace formula have analogues in Connes' Noncommutative Geometry. This paper shows, on several significant examples of Noncommutative Geometry, that Connes' definition of geodesic flow leads to statements analogous to the classical trace formula of Chazarain, Duistermaat, and Guillemin. 1998 Academic Press 0. INTRODUCTION Let X be a compact connected smooth Riemannian manifold endowed with a Hermitian Clifford module E [B-G-V, Chap. 3.3]. Let D be a selfadjoint Dirac type operator acting on the L 2 -sections of E and denote by _ 1

t the geodesic flow on the unitary cotangent bundle S*X of X. Let f # C (X), viewed as a multiplication operator on L 2 (X). For any fixed t # R, f e it |D| is not trace class on L 2 (X); yet one can define a distribution Z f on R by t [ Z f (t)=Trace[ fe it |D| ].