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


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?



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