If $X$ is a smooth manifold on which a Lie group $G$ acts properly and cocompactly (meaning $X/G$ is compact), then one can find a compactly support cut-off function $c:X\rightarrow\mathbb{R}$ such that
$$\int_G c(g^{-1}x)\,dg = 1$$
for all $x\in X$. Then one has a canonical class $[p]\in C_0(X)\rtimes G$, where $p$ is the function in $C_c(G,C_0(X))$ defined by
$$p(g)(x)=c^{1/2}(x)c^{1/2}(g^{-1}x).$$
Now suppose $X/G$ is non-compact. Then cut-off functions $c$ also exist, but the function $p$ defined as above is no longer guaranteed to be in $C_c(G,C_0(X))$, hence one does not automatically have a class $[p]$ as in the cocompact situation.
However, it is not clear to me whether it is possible to define a cut-off function such that $p$ still belongs to the larger algebra $C_0(X)\rtimes G$, although I have heard that this algebra has no projections at all when $X/G$ is non-compact.
Question: Is it true that there are no projections in $C_0(X)\rtimes G$, and how does one show it?
