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

Tingley's problem through the facial structure of operator algebras