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.



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 $A\to B(H)$.

  • $\begingroup$ Can this somehow be used to show that any $*$-homomorphism of $C^*$-algebras $A\rightarrow B$ extends to a $*$-homomorphism $M(A)\rightarrow M(B)$? (I'm slightly worried about the non-degeneracy requirement in Davidson.) $\endgroup$ – ougoah Nov 2 '18 at 2:34
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    $\begingroup$ As far as I can tell, you usually require the $*$-homomorphism to be onto for that. In that case, it looks like the above argument would indeed work. $\endgroup$ – Martin Argerami Nov 2 '18 at 3:51
  • $\begingroup$ If the representation $\pi:A→B(H)$ is degenerate,can we also get the unique extension $\tilde{\pi}:M(A)→B(H)$ by defining $\tilde{\pi}(\tilde a)=lim\pi(a)e_n$,where $\{e_n\}$ is a.u for A. $\endgroup$ – math112358 Apr 27 at 22:23
  • $\begingroup$ You will never get uniqueness of an extension of a degenerate representation; that's precisely the problem. As for the definition, I see no reason for your limit to converge. $\endgroup$ – Martin Argerami Apr 27 at 22:34
  • $\begingroup$ math.stackexchange.com/questions/2940320/… answered the question before.In Blackdar's book,there also states the non-degeneracy of the representation. $\endgroup$ – math112358 Apr 27 at 22:47

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