Order type Tingley's problem for type I finite von Neumann algebras

Let M and N be von Neumann algebras, S(M)+ be the collection of all positive norm one elements in M, and PM be the projection lattice of M. Let phi : PM -> PN be a metric preserving order isomorphism and Lambda : S(M)+ -> S(N)+ be a bijective isometry. When both M and N are type I finite, we establish that the map phi extends to a Jordan *-isomorphism from M onto N. On the other hand, if M and N are of the form (R) k0n=1Mn, where Mn is either zero or a von Neumann algebra of type In, then the map Lambda extends to a Jordan *-isomorphism from M onto N. On our way, we also verify that when M and N are general von Neumann algebras, the map Lambda extends to a Jordan *-isomorphism if and only if Lambda|PM\{0} is a bi-orthogonality preserving bijection from PM \ {0} onto PN \ {0}.(c) 2023 Elsevier Inc. All rights reserved.