Uniformly continuous functions on non-Hausdorff groupoids

The purpose of this paper is to study the notion of uniform continuity introduced in [M. Buneci, Haar systems and homomorphism on groupoids, Operator algebras and mathematical physics, 35-49, Theta, Bucharest, 2003]. For a locally compact (not necessarily Hausdorff) groupoid endowed with pre-Haar systems, we prove that the space of bounded compactly supported functions which are left and right uniformly continuous on fibres can be made into a *-algebra and endowed with a (reduced) C*-norm. The advantage of working with uniformly continuous on fibres functions is the fact that even if the groupoid does not admit a continuous Haar system, various C*-algebras can be associated with it.