Isometries between groups of invertible elements in Banach algebras

We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then T(1)(-1)T is an isometrical group isomorphism. In particular, T(1)(-1)T extends to an isometrical real algebra isomorphism from A onto B.