Characterizations of Jordan \(\dagger\)-skew multiplicative maps on operator algebras of indefinite inner product spaces (Q816625)

The authors present structural results for Jordan *-type maps (without assuming linearity) on operator algebras over complete indefinite inner product spaces. Let \(H\) and \(K\) be real or complex complete indefinite inner product spaces with \(\dagger\) standing for the adjoint operation related to the underlying indefinite structures. It is proved that if \(\Phi:B(H)\to B(K)\) is a bijective map (linearity is not assumed) which satisfies \[ \Phi(AB^\dagger +B^\dagger A)=\Phi(A)\Phi(B)^\dagger+\Phi(B)^\dagger \Phi(A) \] for every pair \(A,B\in B(H)\), then \(\Phi\) is automatically additive. Next, the structure of \(\Phi\) is completely described. It is shown that there exist a nonzero real number \(c\) and a continuous invertible linear or conjugate-linear operator \(U\) with \(U^\dagger U=c^{-1}I\) such that either \(\Phi(A)=cUAU^\dagger\) for all \(A\in B(H)\) or \(\Phi(A)=cUA^\dagger U^\dagger\) for all \(A\in B(H)\). This means that \(\Phi\) is either a linear \(\dagger\)-isomorphism, or a linear \(\dagger\)-anti-isomorphism, or a conjugate-linear \(\dagger\)-isomorphism, or a conjugate-linear \(\dagger\)-anti-isomorphism. A similar description of all bijective maps \(\Phi:B(H)\to B(K)\) satisfying \[ \Phi(AB^\dagger A)=\Phi(A)\Phi(B)^\dagger \Phi(A) \] for every pair \(A,B\in B(H)\) is also presented. Finally, some applications of these results to the Hilbert space case are given.