Banach-Lie algebroids associated to the groupoid of partially invertible elements of a W∗-algebra

In the paper we study the algebroid A(M) of the groupoid G(M)⇉L(M) of partially invertible elements over the lattice L(M) of orthogonal projections of a W∗-algebra M. In particular the complex Banach manifold structure of these objects is investigated. The expressions on the algebroid Lie brackets for A(M) and related algebroids are given in noncommutative operator coordinates in the explicit way. We also describe structure of the groupoid of partial isometries U(M)⇉L(M) and the frame groupoid GlinAp0(M)⇉Lp0(M) as well as their algebroids.