On geometric structure of generalized projections in C~*-algebras

Let H be a Hilbert space and A ■ B(H) be a C~*-subalgebra. This paper is devoted to studying the set gP of generalized projections in A from a differential geometric point of view, and mainly focuses on geodesic curves. We prove that the space gP is a C~∞ Banach submanifold of A, and a homogeneous reductive space under the action of Banach Lie group U_A of A. Moreover, we compute the geodesics of gP in a standard fashion, and prove that any generalized projection in a prescribed neighbourhood of p∈gP can be joined with p by a unique geodesic curve in gP.