University of TehranJournal of Sciences, Islamic Republic of Iran1016-110428220170401C*-algebras on r-discrete Abelian Groupoids16917460753ENH.MyrnouriDepartment of Mathematics, Faculty of Mathematical Sciences, Lahijan Branch of Islamic Azad University, Guilan, Islamic Republic of IranJournal Article20160727<span>We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study the norm-one eigenvectors of corresponding full C*-algebra instead of the multiplicative functionals (spectrum) which have norm-one in the abelian group case.</span><br /> <span> </span>https://jsciences.ut.ac.ir/article_60753_501ae04fc0e5201548f00cf39abfe65e.pdf