  The classical theory of Sobolev towers allows for the construction of an
infinite ascending chain of extrapolation spaces and an infinite descending
chain of interpolation spaces associated with a given $C_0$-semigroup on a
Banach space. In this note we first generalize the latter to the case of a
strongly continuous and exponentially equicontinuous semigroup on a complete
locally convex space. As a new concept - even for $C_0$-semigroups on Banach
spaces - we then define a universal extrapolation space as the completion of
the inductive limit of the ascending chain. Under mild assumptions we show that
the semigroup extends to this space and that it is generated by an automorphism
of the latter. Dually, we define a universal interpolation space as the
projective limit of the descending chain. We show that the restriction of the
initial semigroup to this space is again a semigroup and always has an
automorphism as generator.
