Geometry of the projective unitary group of a C*-algebra

Abstract

Let A be a C∗-algebra with a faithful state ϕ. It is proved thatthe projective unitary group P UA of A,P UA = UA/T.1,(UA denotes the unitary group of A) is a C∞-submanifold of the Banach spaceBs(A) of bounded operators acting in A, which are symmetric for the ϕ-innerproduct, and are usually called symmetrizable linear operators in A.A quotient Finsler metric is introduced in P UA, following the theory ofhomogeneous spaces of the unitary group of a C∗-algebra. Curves of minimallength with any given initial conditions are exhibited. Also it is proved that ifA is a von Neumann algebra (or more generally, an algebra where the unitarygroup is exponential) two elements in P UA can be joined by a minimal curve.In the case when A is a von Neumann algebra with a finite trace, theseminimality results hold for the quotient of the metric induced by the p-normof the trace (p ≥ 2), which metrizes the strong operator topology of P UA.