GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY

Abstract

If [Formula: see text] is a type II1 von Neumann algebra with a faithful trace τ, we consider the set [Formula: see text] of self-adjoint projections of [Formula: see text] as a subset of the Hilbert space [Formula: see text]. We prove that though it is not a differentiable submanifold, the geodesics of the natural Levi–Civita connection given by the trace have minimal length. More precisely: the curves of the form γ(t) = eitxpe-itx with x* = x, pxp = (1 - p)x(1 - p) = 0 have minimal length when measured in the Hilbert space norm of [Formula: see text], provided that the operator norm ‖x‖ is less or equal than π/2. Moreover, any two projections which are unitary equivalent are joined by at least one such minimal geodesic, and only unitary equivalent projections can be joined by a smooth curve. Finally, we prove that these geodesics have also minimal length if one measures them with the Schatten k-norms of τ, ‖x‖k = τ((x* x)k/2)1/k, for all k ∈ ℝ, k ≥ 0. We also characterize curves of unitaries which have minimal length with these k-norms.