Topological fundamental groupoid. III. Haar systems on the fundamental groupoid

2023/05/11 by Holkar, Rohit Dilip, Hossain, Md Amir
#22A22 #28C10 #28C15 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2305.06926

Abstract

Let X be a path connected, locally path connected and semilocally simply connected space; let X be its universal cover. We discuss the existence and description of a Haar system on the fundamental groupoid Π₁(X) of X. The existence of a Haar system on Π₁(X) is justified when X is a second countable, locally compact and Hausdorff. We provide equivalent criteria for the existence of the Haar system on a locally compact (locally Hausdorff) fundamental groupoid in terms of certain measures on X and X. C^*(Π₁(X)) is described using a result of Muhly, Renault and Williams. Finally, two formulae for the Haar system on Π₁(X) in terms of measures on X or X are given.

Related

• Haar Systems on Equivalent Groupoids
• Topological Fundamental Groupoid. II. An action category of the fundamental groupoid
• Topological fundamental groupoid. I
• Induced Representation of Topological groupoids
• Topological amenability is a Borel property