2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157893With 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.A few corrections made; the revised and expanded version of this paper with the new title has been accepted in Illinois J. MathOperator Algebrastext