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Take $X$ to be a locally compact Hausdorf space, $C_b(X)$ to be bounded continuous functions and $C_0(X)$ to be bounded continuous functions that vanish at infinity. If $X$ is compact these notions coincide but in the case of say $\mathbb{R}$ the latter is a subalgebra of the former.

As far as I can tell, both of these are abelian C* algebras. However, $C_0(X)$ seems to be preferred as the prototypical example.

From wikipedia on C* algebras:

Another important class of non-Hilbert C*-algebras includes the algebra $C_0(X)$ of complex-valued continuous functions on X that vanish at infinity, where X is a locally compact Hausdorff space.

From https://math.dartmouth.edu/~dana/bookspapers/cstar.pdf

The Banach algebras $C_0(X)$ (for X locally compact Hausdorff) will be our favorite example of a commutative Banach algebra.

Is this simply because $C_b(X)$ is a bit more unruly? It seems like the "goal" of this it to show via the Gelfand Representation that all abelian C* Algebras are in some sense functions on a compact space, is the point that $C_0(X)$ is "the same as" $C(X\cup{\infty})$ (the one point compactification), but the same is not true for $C_b(X)$ ?

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  • $\begingroup$ I think one reason might be also that often (say $X$ $\sigma$-compact, Hausdorff) $C_b(X)$ may be not separable but $C_0(X)$ will be separable. For example consider $X= \mathbb R$. $\endgroup$ 7 hours ago

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Well, there is one very simple reason: every commutative $C^*$-algebra is isomorphic to $C_0(X)$ for some locally compact Hausdorff space $X$, but not every commutative $C^*$-algebra is isomorphic to $C_b(X)$ for some locally compact Hausdorff space $X$. So the form $C_0(X)$ is not just "prototypical"; it captures every commutative $C^*$-algebra (up to isomorphism).

You can quickly see that not every commutative $C^*$-algebra is isomorphic to an algebra of the form $C_b(X)$ because $C_b(X)$ is always unital. However, this is the only obstruction: every unital commutative $C^*$-algebra is isomorphic to an algebra of the form $C_b(X)$. However, if you restrict to unital algebras, you can do even better: every unital commutative $C^*$-algebra is isomorphic to an algebra of the form $C(X)$ where $X$ is a compact Hausdorff space. Moreover, this $X$ is unique up to homeomorphism and this even extends to a contravariant equivalence of categories between the category of unital commutative $C^*$-algebras and the category of compact Hausdorff spaces. So in the unital case it is better to consider such $C(X)$ as the "prototypical" example.

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  • $\begingroup$ Ah so in a sense C(X) makes the strongest claim about unital C* algebras, and C_0(X) makes the strongest claim about non-unital C* algebras, so C_b just doesn't add much. $\endgroup$
    – lukemassa
    Jun 17, 2021 at 20:34
  • $\begingroup$ Unless I'm mistaken, in general, $C_b(X)$ is (perhaps somewhat tautologically, depending on your definition of $\beta X$) naturally isomorphic to $C(\beta X)$, so the classes of $C_b(X)$ spaces for arbitrary topological $X$ and $C(Y)$ spaces for compact $Y$ are the same (in the language of category theory, we have an equivalence of categories). $\endgroup$
    – tomasz
    Dec 10, 2021 at 19:47

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