Algebroids associated to the groupoid of partially invertible elements of a $W^*$-algebra

Abstract: In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W<sup>*$-algebra. In particular the complex analytic manifold structure of these objects is investigated. The expressions on the Lie brackets for A and related algebroids are given in noncommutative operator coordinates in the explicit way. We also prove statements describing structure of the groupoid of partial isometries and the frame groupoid of A as well as the structure of their algebroids.