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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?

Calculix
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  • Do you know the answer for non-separable, non-abelian compact groups (I don't)? – Plop Oct 21 '21 at 11:13
  • @Plop No, unfortunately not. Everything I know is mentioned in the post. – Calculix Oct 21 '21 at 11:15
  • And is there a chance that you missed a line saying "from now on, we assume $G$ to be second countable" somewhere in the book :)? (even though your question is legitimate) – Plop Oct 21 '21 at 11:16
  • @Plop I checked again, just to be sure, but no. Moreover, he mentions the existence of this ONB to sketch a proof for a result which is known to be true for general locally compact groups. – Calculix Oct 21 '21 at 11:23
  • Hmmm, is it in the proof of Theorem II.10.4.3. on the author's website? Well, if I had read the book, my first reaction would probably have been: "if such a claim is made without further explanation in the non-separable case, it's probably because the author has forgotten to explicitly add a separability hypothesis". – Plop Oct 21 '21 at 11:34
  • Oh, I will take a look at $\mathbb{R}$ with the discrete topology. EDIT: Hahaha, ok, the claim is obvious for discrete groups... – Plop Oct 21 '21 at 11:36
  • @Plop It is indeed Theorem II.10.4.3. But it is known that this theorem is also known for non-second-countable groups. So I don't think he forgot to add further hypotheses. – Calculix Oct 21 '21 at 11:42

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