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.