Rectifiable diameters of the Grassmann spaces of certain von Neumann algebras and C*-algebras

We prove that any two homotopic projections in certain C*-algebras can be connected by a rectifiable path of projections whose length is bounded by a universal constant. In comparison, N.C. Phillips (1992) proved that there are C*-algebras in which such a universal constant does not exist. Our techniques are to estimate the number of symmetries needed to conjugate any two homotopic projections and to factor a unitary in the identity path component as a product of a limited number of symmetries.