An Index Theorem for Fredholm Operators via the Unitary Conjugation Groupoid

Abstract

In a previous paper we introduced the unitary conjugation groupoid associated to any unital separable Type I C*-algebra. This groupoid encodes the representation-theoretic structure of the algebra through the action of its unitary group on the characters of commutative subalgebras and admits a canonical embedding of the algebra into its groupoid C*-algebra. In this paper we apply this framework to two fundamental operator algebras: the algebra of all bounded operators on a separable Hilbert space and the unitization of the compact operators. For any Fredholm operator in these algebras we construct a natural equivariant KK-theory class using the phase of the operator. Applying Kasparov descent for groupoids produces a class in the K-theory of the associated groupoid C*-algebra. Using the Morita equivalences established in the previous work, this descended class can be identified with the symbol class of the operator. We prove that composing this class with the boundary map of the Calkin extension recovers the classical Fredholm index. In particular, the construction yields index minus one for the unilateral shift and index zero for compact perturbations of the identity. This provides a groupoid equivariant formulation of the Fredholm index and connects the unitary conjugation groupoid with classical operator index theory. This perspective suggests a general approach to index theory through equivariant K-theory of groupoids associated with operator algebras.