An application of Dye's theorem on projection lattices to orthogonally decomposable isomorphisms
The predual \(M_*\) of a von Neumann algebra M is an orthogonal decomposable complex Banach space in the sense of \textit{S. Yamamoto} [Bull. Austral. Math. Soc. 30, 357-380 (1984; Zbl 0578.46008)]. The relevant orthogonal decomposition of a selfadjoint \(\omega \in M_*\) is given by the Jordan decomposition, i.e. there exist two unique positive elements \(\omega_{\pm}\in M^+_*\) such that \(\omega =\omega_+- \omega_-\) and the support projections supp \(\omega_{\pm}\) are orthogonal. The author investigates orthogonally decomposable homomorphisms of \(M_*\), i.e. order preserving homomorphisms which also preserves the orthogonal decomposability of \(M_*\). His main result is the following theorem: 1.1 Let \(M_ i\) \((i=1,2)\) be \(W^*\)-algebras with one of them of type I having no direct summands of type \(I_ 2\) in the large. 1.2 A map \(\alpha\) is orthogonally decomposable isomorphism of \((M_ 1)_*\) onto \((M_ 2)_*\), if and only if there exists a positive central element \(\lambda_{\alpha}\) of \(M_ 2\) with a bounded inverse and a Jordan isomorphism \(\beta_{\alpha}\) of \(M_ 2\) onto \(M_ 1\) such that \[ \alpha \omega =\omega \beta_{\alpha}\lambda_{\alpha}\quad (\omega \in (M_ 1)_*). \] Specializing this main theorem to a norm preserving \(\alpha\), the author obtains a nice generalization of the theorem of \textit{E. P. Wigner} [Group theory and its applications to quantum theory of atomic spectra, Academic Press (1959; Zbl 0085.379)] to type I von Neumann algebras with a possibly nontrivial center: 2.1 Assume that 1.1 holds, then a bijective map \(\alpha\) of \((M_ 1)_*\) onto \((M_ 2)_*\) is affine and preserves orthogonality of states, if and only if there exists a Jordan isomorphism \(\beta_{\alpha}\) of \(M_ 2\) onto \(M_ 1\) such that \[ \alpha \omega =\omega \beta_{\alpha}\quad (\omega \in (M_ 1)^+_{*_ 1}). \] The authors' main theorem is related to an isomorphism between the lattices \(P(M_ 1)\) and \(P(M_ 2)\) of orthogonal projections in \(M_ 1\) and \(M_ 2:\) 3.1 If \(\alpha\) is an orthogonally decomposable isomorphism of \((M_ 1)_*\) onto \((M_ 2)_*\), then there exists a unique isomorphism \(\gamma_{\alpha}\) of the orthocomplemented lattice \(P(M_ 1)\) onto \(P(M_ 2)\) such that \[ \gamma_{\alpha}(\sup p \omega)=\sup p(\alpha \omega)\quad (\omega \in (M_ 1)^+_*). \] 3.2 Assume that \(M_ 1\) does not have any type \(I_ 2\) direct summand in the large. If \(\gamma\) is an isomorphism of the complete orthocomplemented lattice \(P(M_ 1)\) onto \(P(M_ 2)\), then there exists a unique orthogonally decomposable isomorphism \(\alpha_{\gamma}\) of \((M_ 1)_*\) onto \((M_ 2)_*\) such that \[ (\gamma_{\alpha}(\omega))(\gamma p)=\omega (p)\quad (p\in P(M_ 1),\quad \omega \in (M_ 1)_*). \] By an application of a theorem of \textit{H. A. Dye} [Ann. Math. 61, 73-89 (1955; Zbl 0064.110)] on projection lattices, the author shows that an orthogonally decomposable isomorphism is implemented by a Jordan isomorphism of M, if the von Neumann algebra M is of type I without direct summands of type \(I_ 2\) in the large.
scientific article; zbMATH DE number 4140553