Given any infinite dimensional unital $C^*$ algebra $A$,does there must exist a non-unital $C^*$ algebra $B$ such that the multiplier algebra $M(B)$ of $B$ is $A$?
2026-03-27 07:18:46.1774595926
multiplier algebra of a non-unital $C^*$ algebra
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No. The multiplier algebra of $M(B)$ of $B$ has $B$ as an essential ideal. So any simple $A$ will be a counterexample.