Automorphisms on algebras of operator-valued Lipschitz maps (Q2915424)

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,NEWLINE 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.