Automorphisms on algebras of operator-valued Lipschitz maps

Let Lip(X, B(H) and lip(alpha)(X, B(H) (0 < alpha < 1) be the big and little Banach *-algebras of B(H)-valued Lipschitz maps on X, respectively, where X is a compact metric space and B(H) is the C*-algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space H. We prove that every linear bijective map that preserves zero products in both directions from Lip(X, B(H)) or lip(alpha)(X, B(H)) onto itself is biseparating. We give a Banach-Stone type description for the *-automorphisms on such Lipschitz *-algebras, and we show that the algebraic reflexivity of the *-automorphism groups of Lip(X, B(H)) and lip(alpha)(X, B(H)) holds for H separable.