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Maps preserving semi-Fredholm operators on Hilbert \(C^*\)-modules. (English) Zbl 1173.46037

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 M.Mbekhta and P.Šemrl [Linear Multilinear Algebra 57, No.1, 55–64 (2009; Zbl 1201.47015)].

MSC:

46L08 \(C^*\)-modules
46L05 General theory of \(C^*\)-algebras
47A53 (Semi-) Fredholm operators; index theories

Citations:

Zbl 1201.47015
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References:

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