Abstract
We study morphisms of the generalized quantum logic of tripotents in JBW\(^*\)-triples and von Neumann algebras. Especially, we establish a generalization of celebrated Dye’s theorem on orthoisomorphisms between von Neumann lattices to this new context. We show the existence of a one-to-one correspondence between the following maps: (1) quantum logic morphisms between the posets of tripotents preserving reflection \(u\rightarrow - u\) (2) maps between triples that preserve tripotents and are real linear on sets of elements with bounded range tripotents. In a more general description we show that quantum logic morphisms on structure of tripotents are given by a family of Jordan *-homomorphisms on Peirce 2-subspaces. By examples we demonstrate optimality of the results. Besides we show that the set of partial isometries with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.
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This work was supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778 The author would like to thank to the referee for many helpful suggestions and careful reading of the manuscript.
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Communicated by Ngai-Ching Wong.
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Hamhalter, J. Dye’s theorem for tripotents in von Neumann algebras and JBW\(^*\)-triples. Banach J. Math. Anal. 15, 49 (2021). https://doi.org/10.1007/s43037-021-00134-w
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DOI: https://doi.org/10.1007/s43037-021-00134-w