Zero product preserving maps on Banach algebras of Lipschitz functions

Let ( K , d ) be a non-empty, compact metric space and α ∈ ] 0 , 1 [ . Let A be either lip α ( K ) or Lip α ( K ) and let B be a commutative unital Banach algebra. We show that every continuous linear map T : A → B with the property that T ( f ) T ( g ) = 0 whenever f , g ∈ A are such that f g = 0 is of the form T = w Φ for some invertible element w in B and some continuous epimorphism Φ : A → B .