  With a minor change made in the previous construction we observe that any
reduced HNN extension is precisely a compressed algebra of a certain reduced
amalgamated free product in both the von Neumann algebra and the $C^*$-algebra
settings. It is also pointed out that the same fact holds true even for
universal HNN extensions of C$^*$-algebras. We apply the observation to the
questions of factoriality and type classification of HNN extensions of von
Neumann algebras and also those of simplicity and $K$-theory of (reduced or
universal) HNN extensions of $C^*$-algebras.
