\(L^{1}\)-algebra of a locally compact groupoid (Q555246)

Summary: For a locally compact groupoid \(G\) with a fixed Haar system \(\lambda\) and quasi-invariant measure \(\mu\), we introduce the notion of \(\lambda\)-measurability and construct the space \(L^{1}(G, \lambda, \mu)\) of absolutely integrable functions on \(G\) and show that it is a Banach \(\ast\)-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 a certain class of nondegenerate representations of \(L^{1}(G, \lambda, \mu)\).