Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book
"Operator algebras, theory of $C^{*}$-algebras and von Neumann algebras"
written by Bruce Blackadar it is claimed (without proof) that $L^2(G)$ admits an orthonormal basis contained in $C_c(G)$.
I didn't immediately see why this is true. So I started to look for an argument and encountered an MSE-post, which shows that it is not always possible to find an orthonormal basis for a non-separable Hilbert space in a given dense subspace.
I know that Blackadar's claim is true in the following cases:
- If $G$ is second countable, then $L^{2}(G)$ is separable. So one can use the Gram-Schmidt procedure to find an orthonormal basis in $C_{c}(G)$.
- If $G$ is compact and abelian, then $\widehat{G}$ (= Pontryagin dual) is an orthonormal basis for $L^{2}(G)$. Note that $\widehat{G}$ can be viewed as a subset of $C_{c}(G)$ in this case.
But does anyone know why Blackadar's claim is true (or false) for general $G$? Or does anyone know a reference for this?
