Maps preserving semi-Fredholm operators on Hilbert \(C^*\)-modules
Let \(M\) denote a Hilbert \(C^*\)-module over a \(C^*\)-algebra \(A\). Let \(L(M)\) and \(K(M)\) be the \(C^*\)-algebras of all adjointable linear operators and compact operators on \(M\), respectively. The authors prove that if \(\varphi: L(M) \to L(M)\) is a linear map preserving semi-\(A\)-Fredholm operators in both directions, \(\varphi(I) = I\), and \(\varphi\) is surjective up to compact operators, then \(\varphi(K(M))\subseteq K(M)\). Furthermore, they show that the induced map \(\widetilde{\varphi}: L(M)/K(M) \to L(M)/K(M)\) is either an automorphism or an anti-automorphism, if one of the following conditions holds: (i) \(M\) is a \(K(H)\)-Hilbert module; (ii) \(M\) is a Hilbert \(C^*\)-module such that \(L(M)\) has the FS-property (i.e., the set of self-adjoint elements of \(L(M)\) being of finite spectrum is norm dense in the set of self-adjoint elements) and \(K(M)\) has a prime corona algebra; (iii) \(A\) is a \(W^*\)-algebra and \(M\) is a self-dual Hilbert module over \(A\) such that \(K(M)\) has a prime corona algebra. These results generalize Theorem 1.2 of \textit{M.\,Mbekhta} and \textit{P.\,Šemrl} [Linear Multilinear Algebra 57, No.\,1, 55--64 (2009; Zbl 1201.47015)].
scientific article; zbMATH DE number 5555219
scientific article