  We show that if $T$ is an isometry (as metric spaces) from an open subgroup
of the invertible group $A^{-1}$ of a unital Banach algebra $A$ onto an open
subgroup of the invertible group $B^{-1}$ of a unital Banach algebra $B$, then
$T$ is extended to a real-linear isometry up to translation between these
Banach algebras. We consider multiplicativity or unti-multiplicativity of the
isometry. Note that a unital linear isometry between unital semisimple
commutative Banach algebra need be multiplicative. On the other hand, we show
that if $A$ is commutative and $A$ or $B$ are semisimple, then $(T(e_A))^{-1}T$
is extended to a isometrical real algebra isomorphism from $A$ onto $B$. In
particular, $A^{-1}$ is isometric as a metric space to $B^{-1}$ if and only if
they are isometrically isomorphic to each other as metrizable groups if and
only if $A$ is isometrically isomorphic to $B$ as a real Banach algebra; it is
compared by the example of \.Zelazko concerning on non-isomorphic Banach
algebras with homeomorphically isomorphic invertible groups. Maps between
standard operator algebras are also investigated.
