  Tingley's problem asks whether every surjective isometry between the unit
spheres of two Banach spaces admits an extension to a real linear surjective
isometry between the whole spaces. In this paper, we give an affirmative answer
to Tingley's problem when both spaces are preduals of von Neumann algebras, the
spaces of self-adjoint operators in von Neumann algebras or the spaces of
self-adjoint normal functionals on von Neumann algebras. We also show that
every surjective isometry between the unit spheres of unital C$^*$-algebras
restricts to a bijection between their unitary groups. In addition, we show
that every surjective isometry between the normal state spaces or the normal
quasi-state spaces of two von Neumann algebras extends to a linear surjective
isometry.
