representations of Banach *-algebras are continuous

Theorem - Let $\pi :𝒜⟶ℬ(H)$ be a representation (http://planetmath.org/BanachAlgebraRepresentation) of a Banach *-algebra $𝒜$ on a Hilbert space $H$. Then $\pi$ is bounded (http://planetmath.org/BoundedOperator) and $\parallel \pi \parallel \le 1$.