The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding

Abstract

This paper introduces a canonical Polish groupoid associated to any separable unital C*-algebra, termed the unitary conjugation groupoid. It is defined as the semidirect product of the algebra's dual space by its unitary group, acting by conjugation. Classical groupoid models for C*-algebras typically require additional structure such as a Cartan subalgebra and rely on the locally compact Hausdorff framework. In contrast, our construction is entirely canonical but forces a paradigm shift: the natural topologies on the dual space and the unitary group are not locally compact. To address this, we equip the dual space with a Polish topology derived from the weak-star topology on pure states and the unitary group with the strong operator topology. This yields a Polish groupoid admitting a continuous Haar system. We prove that the associated reduced groupoid C*-algebra is Morita equivalent to the original algebra tensored with the compact operators, establishing that the groupoid encodes the K-theory of the algebra. A key feature is a canonical diagonal embedding of the original algebra into the groupoid C*-algebra, which is a unital injective star-homomorphism for separable Type I C*-algebras. We characterize commutativity via this embedding and establish functoriality of the construction under appropriate star-homomorphisms. The theory is illustrated with detailed computations for three fundamental classes: finite-dimensional matrix algebras, commutative algebras over compact metrizable spaces, and the unitized compact operators. These examples demonstrate that our general constructions reduce to familiar objects. We also discuss limitations by analyzing the irrational rotation algebra, a non-Type I algebra not covered by our construction, highlighting directions for future research.