Table of Contents
ISRN Algebra
Volume 2011, Article ID 856709, 17 pages
http://dx.doi.org/10.5402/2011/856709
Research Article

𝐿 1-Algebra of a Locally Compact Groupoid

1Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran
2Faculty of Mathematical Sciences and Computer, Tarbiat Moalem University, 50 Taleghani Avenue, Tehran, Iran

Received 12 May 2011; Accepted 6 June 2011

Academic Editor: B. Rangipour

Copyright © 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.

Abstract

For a locally compact groupoid 𝐺 with a fixed Haar system πœ† and quasi-invariant measure πœ‡ , we introduce the notion of πœ† -measurability and construct the space 𝐿 1( 𝐺 , πœ† , πœ‡ ) of absolutely integrable functions on 𝐺 and show that it is a Banach βˆ— -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 , πœ† , πœ‡ ).