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Automorphisms on algebras of operator-valued Lipschitz maps. (English) Zbl 1289.47163

The authors consider the automorphisms of big and little operator-valued Lipschitz functions Lip\((X, B(H))\) and \(\mathrm{lip}_{\alpha}(X, B(H))\), that is, Lipschitz functions from a compact metric space \(X\) to the \(C^*\)-algebra \(B(H)\) of all bounded and linear operators on a Hilbert space \(H\). They prove that every linear bijective map from one of these algebras onto itself that preserves zero products in both directions is biseparating. A Banach-Stone-type description for the *-automorphisms on such Lipschitz *-algebras is given. If \(H\) is separable, the authors prove the algebraic reflexivity of the *-automorphism groups of the considered Lipschitz algebras.

MSC:

47L10 Algebras of operators on Banach spaces and other topological linear spaces
47B48 Linear operators on Banach algebras
46E40 Spaces of vector- and operator-valued functions
46J10 Banach algebras of continuous functions, function algebras
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