An Algebraic Property of an Isometry between the Groups of Invertible Elements in Banach Algebras

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry from the invertible group of a unital commutative Banach algebra onto that of a unital semisimple Banach algebra, then (T (eA)) −1 T is extended to a surjective isometrical real algebra isomorphism between the two underling algebras.