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

Abstract

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 Φ:PM→PN be a metric preserving order isomorphism and Λ:S(M)+→S(N)+ be a bijective isometry. When both M and N are type I finite, we establish that the map Φ extends to a Jordan ⁎-isomorphism from M onto N. On the other hand, if M and N are of the form ⨁n=1k0Mn, where Mn is either zero or a von Neumann algebra of type In, then the map Λ 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 Λ extends to a Jordan ⁎-isomorphism if and only if Λ|PM∖{0} is a bi-orthogonality preserving bijection from PM∖{0} onto PN∖{0}.