C∗-algebras on r-discrete abelian groupoids

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. © 2017,national research center of medical sciences. all rights reserved.