Automorphisms on algebras of operator-valued Lipschitz maps

Abstract

Let Lip(X; B(H)) and lip (X; B(H)) (0 < < 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 in nite-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 (X; B(H)) onto it- self is biseparating.We give a Banach{Stone type description for the -automorphisms on such Lipschitz -algebras, and we show that the algebraic reexivity of the - automorphism groups of Lip(X; B(H)) and lip (X; B(H)) holds for H separable.