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A Banach lattice $E$ is said to be an AM-space if $$\|\sup\{x,y\}\|=\sup\{\|x\|,\|y\|\}$$ for all positive $x,y\in E$. My question is as follows:

Is every AM-space (which is a $*$-algebra) a $C^*$-algebra?

My intuition is as follows: I think of AM-spaces with unit as a generalization of $C(K)$ and without unit as a generalization of $C_0(L)$. Analogous, intuition for $C^*$-algebras.

Due to this I'm tempted to answer "yes" to the above question, but I want to know if it is actually true.

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