On the normal spectrum of a subnormal operator

In this note we present a new characterization for subnormality which is purely ${C^ \ast }$-algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to prove that ${\sigma _ \bot }(\pi (S)) \subseteq {\sigma _ \bot }(S)$ for any $^ \ast$-representation $\pi$.