Nonlinear maps preserving Jordan \(\ast\)-products. (English) Zbl 1339.47047
Let \(\eta\) be a nonzero complex number. The paper studies a bijective map \(\phi\) between von Neumann algebras \(A\) and \(B\) such that \(\phi(ab + \eta ba^*) = \phi(a)\phi(b) + \eta \phi(b)\phi(a)^*\) for all \(a,b\in A\) (linearity is not assumed). The main result states that if at least one of \(A\), \(B\) has no central abelian projections, then \(\phi\) is a linear \(\ast\)-isomorphism if \(\eta\) is not real, and is the sum of a linear \(\ast\)-isomorphism and a conjugate linear \(\ast\)-isomorphism if \(\eta\) is real.
Reviewer: Matej Brešar (Maribor)
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 46L10 | General theory of von Neumann algebras |
Keywords:
Jordan \(\ast\)-product; von Neumann algebra; isomorphism; maps preserving \(\eta\)-\(\ast\)-productReferences:
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