  We obtain an analog of the compression of angles theorem in symmetric spaces
for Bruhat--Tits buildings of the type $A$.
  More precisely, consider a $p$-adic linear space $V$ and the set $Lat(V)$ of
all lattices in $V$. The complex distance in $Lat(V)$ is a complete system of
invariants of a pair of points of $Lat(V)$ under the action of the complete
linear group. An element of a Nazarov semigroup is a lattice in the duplicated
linear space $V\oplus V$. We investigate behavior of the complex distance under
the action of the Nazarov semigroup on the set $Lat(V)$.
