\({\ast}\)-Jordan-triple multiplicative surjective maps on \(\mathcal B(H)\)
Let \(\mathbb{F}\) a real or complex field and \(\mathbb{F}^{\ast}=\mathbb{F}\backslash\{0\}\). Let \(H\) be a (real or complex) Hilbert space and denote by \(B(H)\) the algebra of all bounded linear operators on \(H\). Recall that the Jordan-triple product is defined in the following way: \(A\circ B=ABA\), where \(A,B\) are (for example) operators in \(B(H)\). Many authors have studied preservers of the Jordan-triple product, i.e., maps \(\phi\) satisfying \(\phi(ABA)=\phi (A)\phi(B)\phi(A)\), as well as maps \(\phi\) satisfying \(ABA=0\) if and only if \(\phi(A)\phi(B)\phi(A)=0\). In this paper, the author studies problems of this kind for the case of another product on \(B(H)\), defined by \(A\ast B=AB^{\ast}A,\) \(A,B\in B(H).\) The paper is divided into three chapters. The main result is as follows: Let \(H\) be an infinite-dimensional Hilbert space and \(G_{1},G_{2}\subseteq B(H)\) be subsets containing \(\mathbb{F} I\) and all rank-one idempotents. Assume that \(\phi: G_{1}\rightarrow G_{2}\) is a surjective map and \(\phi(\mathbb{F} I)=\mathbb{F}I\). Then \[ AB^{\ast}A=0\quad\text{iff\quad}\phi(A)\phi(B)^{\ast}\phi(A)=0 \tag{en} \] for every \(A,B\in G_{1}\), if and only if there exists a function \(\alpha:G_{1}\rightarrow\mathbb{F}^{\ast}\) and a unitary or antiunitary operator \(U:H\rightarrow H\) such that either \(\phi(A)=\alpha(A)UAU^{\ast}\) for all \(A\in G_{1}\), or \(\phi(A)=\alpha(A)UA^{\ast}U^{\ast}\) for all \(A\in G_{1}\). The proof of this result follows ``usual steps'', namely, the author first studies surjective maps which satisfy ({en}) for the special case when \(G_{1}=G_{2}\) is the set of all rank-one idempotent operators from \(B(H)\). He shows that such maps preserve orthogonality, i.e., \(PQ=0=QP,\) and then uses a known result due to \textit{P. Šemrl} [Acta Sci. Math. 71, No. 3-4, 781--819 (2005; Zbl 1111.15002)]. The main result is applied to obtain a characterization of surjective maps \(\phi:B(H)\rightarrow B(H)\), where \(\phi(AB^{\ast}A)=\phi(A)\phi(B)^{\ast}\phi(A)\) for every \(A,B\in B(H)\). Here, \(H\) is again an infinite-dimensional Hilbert space.
scientific article; zbMATH DE number 6143680