Short geodesics of unitaries in the L2 metric

Abstract

Let M be a type II_1 von Neumann algebra, τ a trace in M, and l^2 (M,τ) the GNS Hilbert space of τ. We regard the unitary group U_M as a subset of l^2 (M,τ), and characterize the shortest smooth curves of unitaries joining two fixed unitaries, in the L^2 metric. As a consequence of this we obtain that U_M, though a complete (metric) topological group, is not an embedded riemannian submanifold of l^2 (M,τ)