International Scholarly Research Network ISRN Algebra Volume 2011, Article ID 856709, 17 pages doi:10.5402/2011/856709 Research Article L 1 -Algebra of a Locally Compact Groupoid Massoud Amini, 1 Alireza Medghalchi, 2 and Ahmad Shirinkalam 2 1 Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran 2 Faculty of Mathematical Sciences and Computer, Tarbiat Moalem University, 50 Taleghani Avenue, Tehran, Iran Correspondence should be addressed to Massoud Amini, mamini@modares.ac.ir Received 12 May 2011; Accepted 6 June 2011 Academic Editor: B. Rangipour Copyright q 2011 Massoud Amini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 L 1 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 L 1 G, λ, μ. 1. Introduction and Preliminaries For a locally compact group G with a Haar measure λ, the Banach algebra L 1 G, λ plays a central role in harmonic analysis on G 1. This motivated us to define a similar notion in the case where G is a locally compact groupoid with a fixed Haar system λ and quasi-invariant measure μ. This paper is devoted to the study of such a groupoid L 1 -algebra L 1 G, λ, μ. One may expect that as the group case, there is a full interaction between the properties of G and that of L 1 G, λ, μ. This is not completely true. For instance, unlike the group case, not every nondegenerate representation of L 1 G, λ, μ is integrated form a representation of G. In Section 2, we introduce the appropriate measurability notion used to define L 1 G, λ, μ. Sections 3 and 4 are devoted to the algebra structure of L 1 G, λ, μ and its embedding into MG as a closed ideal. In Section 5, we find the class of nondegenerate representations of L 1 G, λ, μ which could be obtained by integrating a representation of G. We start with some basic definitions. Our main reference for groupoids is the Renault’s book 2. In this paper, we frequently use the following version of Fubini’s theorem for not necessarily σ -finite Radon measures 1, Theorem B.3.3.