Geometry of oblique projections

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projecti ons of A. We study the relationship between P and the spaces of projections P, determined by the different involutions #(a) induced by positive invert ible elements a is an element of A. The maps phi(p) : P --> P-a sending p t o the unique q is an element of P-a with the same range as p and Omega(a) : P-a --> P sending q to the unitary part of the polar decomposition of the symmetry 2q - 1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q, r is an element of A with parallel to q - r parallel to < 1 such that there exists a positive element a is an element of A satisfyin g q, r is an element of P-a. In this case q and r can be joined by a unique short geodesic along the space of idempotents Q of A.