On the index of noncommutative elliptic operators over \(C^*\)-algebras.

*(English. Russian original)*Zbl 1198.19006
Sb. Math. 201, No. 3, 377-417 (2010); translation from Mat. Sb. 201, No. 3, 63-106 (2010).

Let \(\Gamma\) be a discrete group acting by isometries on a smooth compact manifold \(M\), let \(A\) be a \(C^*\)-algebra upon which \(\Gamma\) acts trivially, and define \(\mathcal{H}\) to be the Hilbert space generated by the smooth functions from \(M\) to \(A\). There is a natural notion of pseudodifferential operators on \(\mathcal{H}\), and in this paper the authors produce an index theorem for such operators. The work in this paper is a continuation of the results in the book written by the two authors and V. E. Nazaikinskii [Elliptic theory and noncommutative geometry, Oper. Theory: Adv. Appl. 183, Basel: Birkhäuser (2008; Zbl 1158.58013)]. There are two primary differences between the results in this paper and the ones in the book. First, in this paper the authors model their definition of the Chern character on the one given by M. Karoubi [“Homologie cyclique et \(K\)-théorie”, Astérisque 149 (1987; Zbl 0648.18008)], but instead of employing the universal algebra of noncommutative differential forms, the authors use a more concrete differential graded algebra. Second, instead of only considering pseudodifferential operators over the group algebra \(C^*\Gamma\), the authors work with pseudodifferential operators over more general \(C^*\)-algebras. Among other things, this approach simplifies the proof of the main theorem in the aforementioned book.

Reviewer: Efton Park (Fort Worth)

##### MSC:

19K56 | Index theory |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

58J99 | Partial differential equations on manifolds; differential operators |