DIVISIBLY NORM-PRESERVING MAPS BETWEEN COMMUTATIVE BANACH ALGEBRAS

摘要

Let A and 13 be unital commutative Banach algebras. Suppose that A is semi-simple. Let ρ : A → A and τ : B→B be bijections. If T : A → B is a surjection with, for some α ∈ C{0}, r(T(f)τ(T(g))-α) = r(fρ(g) -α) for all f,g ∈ A, then B is semi-simple and r(T(f)T(g)~(-1) - 1) = r(fg~(-1) - 1) for every f ∈ A and g ∈ A~(-1). As a consequence, T(1) is invertible and T(1)~(-1)T is a real-algebra isomorphism. If, in addition, T(1)~(-1)T(i) = i, then T(1)~(-1)T is a complex-algebra isomorphism. This result unifies and generalizes [3, Theorem 7.4] and [4, Theorem 3.2 and 6.2].