Nonlinear mappings preserving Jordan-type \(\eta\)-\(\ast\)-products. (English) Zbl 1507.47084

Lin, Wenhui

Rocky Mt. J. Math. 52, No. 3, 967-998 (2022).

Summary: Let \(\eta\) be a nonzero complex number. Let \(\mathscr{A}\) and \(\mathscr{B}\) be two von Neumann algebras, one of which has no central abelian projections, and let \(\Phi: \mathscr{A} \to \mathscr{B}\) be a not necessarily linear bijection. For arbitrary elements \(A, B \in\mathscr{A}\), one can define their Jordan \(\eta\)-\(\ast\)-product in the sense of \(A\Diamond_\eta B = AB + \eta BA^\ast\). Let \(p_n (X_1, X_2, \dots, X_n)\) be the polynomial defined by \(n\) indeterminates \(X_1, \dots, X_n\) and their Jordan multiple \(\eta\)-\(\ast\)-product. In this article, it is shown that \(\Phi\) satisfies the condition \[\Phi (p_n (A_1, A_2, \dots, A_n)) = p_n (\Phi (A_1), \Phi (A_2), \dots, \Phi (A_n))\] for all \(A_1, A_2, \dots, A_n \in\mathscr{A}\) if and only if one of the following statements holds true:

(1)

\(\eta \in\mathbb{R}\) and there exists a central projection \(E \in\mathscr{A}\) such that \(\Phi (E)\) is a central projection in \(\mathscr{B}, \Phi |_{\mathscr{A}E} : \mathscr{A}E \to \mathscr{B} \Phi (E)\) is a linear \(\ast\)-isomorphism and \(\Phi|_{\mathscr{A}(I - E)} : \mathscr{A}(I - E) \to \mathscr{B}(I - \Phi (E))\) is a conjugate linear \(\ast\)-isomorphism,

(2)

\(\eta \notin \mathbb{R}\) and \(\Phi\) is a linear \(\ast\)-isomorphism.

MSC:

47B49	Transformers, preservers (linear operators on spaces of linear operators)
46L10	General theory of von Neumann algebras

Keywords:

Jordan-type \(\eta\)-\(\ast\)-product; \(\ast\)-isomorphism; von Neumann algebra

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