Not much progress this weekend. But I had originally written that Bost-Connes type systems where the C* algebra generated by $\mathcal{L}_n / \mathbb{R}^{*}_{+}$ (n=1 for Bost-Connes, n=2 for imaginary quadratic fields.) I got that from the notation $C^{*} (\mathcal{L}_n)$. I've learned a bit more about that symbol.

Its not the C* algebra generated by $\mathcal{L}_n$...its the group C* algebra of $\mathcal{L}_n$, which is apparently the C*-enveloping algebra of $L^1 (\mathcal{L}_n)$ From what my googling and textbook-glancing today has told me, its more related to harmonic analysis and the Haar measure stuff then to the material covered in Averson's Invitation to C* Algebras. The Haar measure is a Borel measure that we can find on any [locally] compact topological group G, according to a theorem in Functional/Harmonic analysis. The notation $L^1 (G)$ is then pretty obvious, its exactly what one would expect from the Lebesque spaces (I think, I need to look into this stuff more.) I have no idea what a C*-enveloping algebra is yet. I'll be looking into all these things from tomorrow onwards.

But its starting to look like $C^{*} (\mathcal{L}_n)$ is a pretty complex object. Its not even obvious to me that $\mathcal{L}_n$ is [locally?] compact! Then again, I'm not worried about that right now. I'm just trying to get the "big picture".

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