# Why is $C_0(X)$ not $C_b(X)$ for $X$ locally compact the "prototypical" abelian $C^*$ Algebra

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.

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 $$C_b(X)$$ ?

• 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 $X= \mathbb R$. 7 hours ago

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.
• 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). Dec 10, 2021 at 19:47