The paraunitary group of a von Neumann algebra

It is proved that the pure paraunitary group over a von Neumann algebra coincides with the structure group of its projection lattice. The structure group of an arbitrary orthomodular lattice (OML) is a group with a right invariant lattice order, and as such it is known to be a complete invariant of the OML. The pure paraunitary group PPU(A)$\mbox{PPU}(\mathcal {A})$ of a von Neumann algebra A$\mathcal {A}$ is a normal subgroup of the paraunitary group PU(A)$\mbox{PU}(\mathcal {A})$ with the group U(A)$\mbox{U}(\mathcal {A})$ of unitaries in A$\mathcal {A}$ as cokernel. By a result of Heunen and Reyes, A$\mathcal {A}$ is determined by the action of U(A)$\mbox{U}(\mathcal {A})$ on PPU(A)$\mbox{PPU}(\mathcal {A})$. In this sense, it follows that the paraunitary group is a complete invariant of any von Neumann algebra.