Resumen de Geometry of the projective unitary group of a C ∗ -algebra

Esteban Andruchow

• Let A be a C∗-algebra with a faithful state φ. It is proved that the projective unitary group PUA of A,PUA=UA/T.1,(UA denotes the unitary group of A) is a C∞-submanifold of the Banach space Bs(A) of bounded operators acting in A, which are symmetric for the φ-inner product, and are usually called symmetrizable linear operators in A. A quotient Finsler metric is introduced in PUA, following the theory of homogeneous spaces of the unitary group of a C∗-algebra. Curves of minimal length with any given initial conditions are exhibited. Also it is proved that if A is a von Neumann algebra (or more generally, an algebra where the unitary group is exponential) two elements in PUA can be joined by a minimal curve.

  In the case when A is a von Neumann algebra with a finite trace, these minimality results hold for the quotient of the metric induced by the p-norm of the trace (p≥2), which metrizes the strong operator topology of PUA.