I have a encountered a technical difficulty while trying to do a certain computation in solving a problem. The problem can be formulated as the following:
Let $X$ and $Y$ be locally compact groups. $\phi: X \rightarrow Y$ is a continuous homomorphism and $f\in C_0(Y)$.
From this information, can we conclude that $f\circ\phi \in C_0(X)$ ?
If this is too strong a conclusion to make, what condition(s) do I need on $X, Y$ or $f$ so that the function $f\circ \phi$ will be included in $C_0(X)$?
I realize I need some kind of boundedness property of $\phi$ to conclude this. But I am not sure if the fact that $\phi$ is a continuous homomorphism automatically implies that.
Any help will be greatly appreciated ! Thank you
I'm assuming $C_0(X)$ means the continuous functions with limit $0$ at $\infty$.
No, it's not true. For example, $\phi: \; (s,t) \to s$ is a continuous homomorphism of locally compact groups $(\mathbb R^2,+)$ to $(\mathbb R,+)$ and $f(x) = 1/(1+x^2)$ is in $C_0(\mathbb R)$, but $f \circ \phi: (x,y) \to 1/(1+x^2)$ is not in $C_0(\mathbb R^2)$.
EDIT: More generally, $f \circ \phi$ can't be in $C_0(X)$ unless $\ker(\phi)$ is compact.