L1-Algebra of a Locally Compact Groupoid

Abstract EN

For a locally compact groupoid G with a fixed Haar system λ and quasi-invariant measure μ, we introduce the notion of λ-measurability and construct the space L1(G, λ, μ) of absolutely integrable functions on G and show that it is a Banach ∗-algebra and a two-sided ideal in the algebra M(G) of complex Radon measures on G.

We find correspondences between representations of G on Hilbert bundles and certain class of nondegenerate representations of L1(G, λ, μ).