Let $A$ be a C$^\*$-algebra, $f$ a representation of $A$, $F$ the universal representation of $A$, and $g=f\circ F^{-1}$. For an ultraweakly continuous linear functional $w$ on $f(A)$, $w\circ g$ is bounded and hence according to a well-known theorem, is ultraweakly continuous linear functional on $F(A)$.
My question: it results that g is ultraweakly continuous. How is its proof?
Thank you for consideration.