A person in my class asked today if $\mathfrak{A}$ is a unital $C^{*}$-algebra and $x,y \in \mathfrak{A}$ commute and are normal, then is it always the case there exists $\alpha \in C^{*}(x,1)$ and $\beta \in C^{*}(y,1)$ such that $C^{*}(x,y,1) = C^{*}(\alpha+\beta,1)$?
I don't think this could be true, as it would seem to insinuate that one could apply Gelfand theory to $C^{*}(x,y,1)$ so that $C^{*}(x,y,1) \hookrightarrow C(\text{sp}_{\mathfrak{A}}(\alpha+\beta))$ and moreover that: $$C^{*}(x,1), C^{*}(y,1) \hookrightarrow C(\text{sp}_{\mathfrak{A}}(\alpha+\beta)).$$ This doesn't seem right to me, as it then seems to imply everything we could know about both $x,y$ can be obtained by looking at some random sum of two elements from the $C^{*}$-algebra they generate.. Especially on the physics side of things, this seems wrong.
Perhaps I'm missing an obvious counter example or perhaps this is 'clearly' true? Any help would be appreciated.