  In the paper we study the algebroid A of the groupoid of partially invertible
elements over the lattice of orthogonal projections of a $W^*$-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.
