0 the Manifold of Finite Rank Projections in the Space L ( H )

Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators in H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. To do it we take the Jordan-Banach triple (or JB *-triple) approach which allows us to define a natural Levi-Civita connection on M by using algebraic tools. We identify the geodesics and the Riemann distance and establish some properties of M. 0 Introduction In this paper we are concerned with the differential geometry of the infinite-dimensional Grassmann manifold M of all projections in Z: = L(H), the space of bounded linear operators z: H → H in a complex Hilbert space H. Grassmann manifolds are a classical object in Differential Geometry and in recent years several authors have considered them in the Banach space setting. Besides the Grassmann structure, a Riemann and a Kähler structure has sometimes been defined even in the infinite-dimensional setting. Let us recall some aspects of the history of the topic that are relevant for our purpose. The study of the manifold of minimal projections in a finite-dimensional simple formally real Jordan algebra was made by U. Hirzebruch in [4], who proved that such a manifold is a compact symmetric Riemann space of rank 1, and that every such a space arises in this way. Later on, Nomura in [13, 14] established similar results for the manifold of fixed finite rank projections in a topologically simple real Jordan-Hilbert algebra. On the other hand, the Grassmann manifold M of all projections in the space Z: = L(H) of bounded linear operators has been discussed by Kaup in [7] and [10]. It is therefore reasonable to ask whether a Riemann structure can always be defined in M and how does it behave when it exists. It is known that M has several connected components M r ⊂ M each of which consists of the projections in L(H) that have a fixed rank r, 1 ≤ r ≤ ∞. We prove that M r admits a Riemann structure if and only if r < ∞ establishing a distinction between the finite and the infinite dimensional cases. We then assume r < ∞ and proceed to discuss the behaviour of the Riemann manifold M r , which looks very much like in the finite-dimensional case. One of the novelties is that we take …