Applications of Connes' geodesic flow to trace formulas in noncommutative geometry (Q1279642)

A noncommutative geometric space is defined by a noncommutative algebra \({\mathcal A}\) of functions over a topological space. Then a Riemannian structure on \({\mathcal A}\) is a spectral triple \(({\mathcal A},H,D)\), where: (A1) \(H\) is a separable Hilbert space, \(D\) is an unbounded self-adjoint operator on \(H\) such that \((1+| D|)^{-1}\) is compact, (A2) there exists a nonnegative real \(d\), called the dimension of the spectral triple \(({\mathcal A},H,D)\) and a positive constant \(C\) such that \({1\over C}\leq {1\over {\text{ log} n}}\sum^n_{j=2} \lambda _j^{-d}\leq C\), where \(\lambda _j\) are the eigenvalues of \(| D| \), (A3) \({\mathcal A}\) is an involutive unital subalgebra \({\mathcal B}^\infty \) stable under holomorphic function calculus in \({\overline{\mathcal A}}\), the norm closure of \({\mathcal A}\) in \(B(H)\). The authors obtain some extensions to the noncommutative case of the properties of the geodesic flow. They study several examples of spectral triples \(({\mathcal A},H,D)\), most of them verifying a certain geometric hypothesis (GH). In the case of each example it is obtained \(H\), then the space \(\chi \) identified with a subset of \({\mathcal P} (\overline{\mathcal A})\) and, then, there are obtained \(S^*{\mathcal A}\) and \({\widetilde \chi}\). The obtained results are analogous to the classical trace formula of Chazarain, Duistermaat and Guillemin.