The manifold of finite rank projections in the algebra L(H) of bounded linear operators

Details The manifold of finite rank projections in the algebra L(H) of bounded linear operators The manifold of finite rank projections in the algebra L(H) of bounded linear operators

2001-10-11

arxiv.org/abs/math/0110115v1

Mathematics

Functional Analysis

17 pages, Latex 2e, to appear in Expositiones Mathematicae

Scientific paper

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V:=L(H). Using the algebraic structure of V, a torsionfree affine connection $\nabla$ (that is invariant under the group of automorphisms of V) is defined on every connected component of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0< r < \infty). We prove that M admits a Riemann structure if and only if M consists of projections that have the same finite rank r or the same finite corank, and in that case $\nabla$ is the Levi-Civita and the K\"ahler connection of M. Moreover, M turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed. Keywords: JBW-algebras, Grassmann manifolds, Riemann manifolds. AMS 2000 Subject Classification: 48G20, 72H51.

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