Burgos, María; Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés Automorphisms on algebras of operator-valued Lipschitz maps. (English) Zbl 1289.47163 Publ. Math. Debr. 81, No. 1-2, 127-144 (2012). 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. Reviewer: Catalin Badea (Villeneuve d’Ascq) Cited in 2 Documents 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 Keywords:algebraic reflexivity; local automorphism; Lipschitz algebra; \(C^*\)-algebra PDFBibTeX XMLCite \textit{M. Burgos} et al., Publ. Math. Debr. 81, No. 1--2, 127--144 (2012; Zbl 1289.47163) Full Text: DOI