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Is there a non unital $C^{*}$ algebra $A$ for which the multiplier algebra $M(A)$ is isomorphic to the minimal unitization $\tilde{A}$?

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For separable $A$, it is never true: in 3.2.12 in Pedersen's C*-Algebras and their Automorphisms, it is shown that for non-unital $A$, $M(A)$ is always non-separable.

In the commutative case, you have $A=C_0(X)$ for some locally compact Hausdorff $X$, and $M(A)=C^b(X)=C(\beta X)$. So in this case the answer is yes for those $X$ with unique compactification; see this question for some details on those spaces.

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  • $\begingroup$ Thank you very much for your answer. What about for $A$ commutative but not necessarily separable? $\endgroup$ – Ali Taghavi Jun 23 '14 at 23:08
  • $\begingroup$ Please see the edit. $\endgroup$ – Martin Argerami Jun 23 '14 at 23:30
  • $\begingroup$ thank you so much for this edited version $\endgroup$ – Ali Taghavi Jun 23 '14 at 23:40

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