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

Functional Analysis