We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C^*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.

翻译：我们证明了着色Rook幺半群$R_n^{(r)}$的代数是有限群胚的代数，即该幺半群由每行每列至多含一个非零元素（$r$次单位根）的$n \times n$矩阵构成，因而具有$C^*$-代数结构。这一新视角通过显式揭示这些幺半群代数在不可约模中的分解，阐明了其表示理论。