Let $A$ be a $C^*$-algebra and $f:A\rightarrow\mathbb{C}$ a $^*$-homomorphism. Does $f$ always extend to a $^*$-homomorphism $\tilde{f}:M(A)\rightarrow\mathbb{C}$, where $M(A)$ is the multiplier algebra of $A$?
I suspect the answer is no, but I'm not sure how to show this. In case the answer is yes, a proof or reference would be greatly appreciated.
Thanks!
Best Answer
The answer is yes. This depends on two facts:
$A$ is an ideal in $M(A)$.
if $J\subset B$ is an ideal, any non-degenerate representation $J\to B(H)$ can be extended to a representation $B\to B(H)$. This is for instance Lemma I.9.14 in Davidson's C$^*$-Algebras By Example. Here we have $J=A$, $B=M(A)$, $H=\mathbb C$.
The argument works for any $*$-homomorphism $A\to B(H)$.