Orthogonality preserving linear maps on \(C^{\ast }\)-algebras with non-zero socles
Let \(\mathcal{A}\) be a \(C^{\ast }\)-algebra and \(a,b\in \mathcal{A}\). If \(ab^{\ast }=b^{\ast }a=0,\) we write \(a\bot b\) and say that \(a\) and \(b\) are orthogonal. A linear map \(T:\mathcal{A\rightarrow B}\) between \(C^{\ast }\)-algebras \(\mathcal{A}\) and \(\mathcal{B}\) is said to be orthogonality preserving if \(a\bot b\) implies \(T(a)\bot T(b)\) for every \(a,b\in \mathcal{A}\). We say that \(T\) is biorthogonality preserving if \(a\bot b\) if and only if \(T(a)\bot T(b)\) for every \(a,b\in \mathcal{A}\). The paper has three chapters. In the first chapter, the author introduces notations and presents known results while in the second chapter some auxiliary results are proved. The first of the two main results which are presented and proved in the third chapter is the following theorem. Let \(\mathcal{A}\) and \(\mathcal{B}\) be \(C^{\ast }\)-algebras and denote by Soc\((\mathcal{A})\) the socle of \(\mathcal{A}\). Suppose that \(\mathcal{A}\) is unital with unit \(1\) and with non-zero socle. Let \(T:\mathcal{A\rightarrow B}\) be an orthogonality preserving bijective linear map with \(\{T(1)\}^{\bot}=\{0\}\), where \(\{T(1)\}^{\bot }=\{y\in \mathcal{B}:y\bot T(1)\}.\) Then \[ T^{-1}(T(1)^{\ast }T(a)-T(a^{\ast })^{\ast }T(1))\text{Soc}(\mathcal{A})=\{0\}, \] and \[ T^{-1}(T(a^{2})T(1)^{\ast }T(1)-T(a)T(a^{\ast })^{\ast }T(1))\text{Soc}(\mathcal{A})=\{0\} \] for every \(a\in \mathcal{A}\). Moreover, if Soc\((\mathcal{A})\) is essential, then \(\mathcal{B}\) is unital, and \(T\) is a Jordan *-isomorphism multiplied by an invertible element. As a direct consequence of this theorem, it follows that every biorthogonality preserving surjective linear map from a unital \(C^{\ast }\)-algebra with essential socle to a \(C^{\ast }\)-algebra is automatically bounded and a multiple of a Jordan *-isomorphism. In conclusion, the author studies orthogonality preserving bijective linear maps \(T\) from a unital \(C^{\ast }\)-algebra with a non-zero socle to a \(C^{\ast }\)-algebra, where \(T(1)\) is a partial isometry. The author shows that, if Soc\((\mathcal{A})\) is essential, then \(T\) is a Jordan *-isomorphism multiplied by a unitary element.
scientific article; zbMATH DE number 6156257