  Denote by $B_n$ the set of complex square matrices of order $n$, whose
Euclidean operator norms are $<1$. Its Shilov boundary is the set $U(n)$ of all
unitary matrices. A holomorphic map $B_m\to B_n$ is inner if it sends $U(m)$ to
$U(n)$. On the other hand we consider a group $U(n+mj)$ and its subgroup $U(j)$
embedded to $U(n+mj)$ in a block-diagonal way ($m$ blocks $U(j)$ and a unit
block of size $n$). For any conjugacy class of $U(n+mj)$ with respect to $U(j)$
we assign a 'characteristic function', which is a rational inner map $B_m\to
B_n$. We show that the class of inner functions, which can be obtained as
'characteristic functions', is closed with respect to natural operations as
pointwise direct sums, pointwise products, compositions, substitutions to
finite-dimensional representations of general linear groups, etc. We also
describe explicitly the corresponding operations on conjugacy classes.
