Fredholm complexes of Hilbert C*-modules

TL;DR

This work extends Fredholm theory to complexes of Hilbert C*-modules, introducing the even Dirac operator as a central unifying object. By developing bounded-transform, adjoint, and graph-norm reformulations, it establishes rigorous equivalences between the Fredholm property of a complex and the Fredholmness of its Dirac and Laplace operators, and defines the Fredholm index in $K_0(\mathcal{A})$ with robust stability under perturbations. The paper proves powerful structural results, including Putinar's functor adaptations to compute the index and a suite of invariance theorems under small, relatively compact, or tensor perturbations, direct sums, and exact sequences. It also provides concrete examples with C*-algebras of compact operators and $\mathcal{A}$-elliptic complexes, illustrating the theory's breadth and applicability to noncommutative settings. Overall, the framework blends noncommutative geometry with operator-theoretic homological methods to generalize Segal-type Euler characteristics to a broad noncommutative context.

Abstract

We investigate complexes of Hilbert C*-modules, which are cochain complexes with (unbounded) regular operators between Hilbert C*-modules as differential maps. In particular, we provide various equivalent characterizations of the Fredholm property for such complexes of Hilbert C*-modules, and we define the Fredholm index taking values in the K-theory group of the C*-algebra. Among other properties of this index, we prove the stability under small or relatively compact perturbations, and we obtain alternative expressions for the index under the existence of a (weak or strong) Hodge decomposition.

Theorems & Definitions (228)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5: Weak Hodge decomposition
• Theorem 1.6: Strong Hodge decomposition
• Theorem 1.7
• Definition 1.8
• Definition 1.9
• Remark 1.10