Existence of tracial states on reduced group C*-algebras

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Let G be a locally compact group. It is not always the case that its reduced C*-algebra C*ᵣ(G) admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when G compactly generated. In particular for G almost connected, or more generally when C*ᵣ(G) is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups for which C*ᵣ(G) does not admit a tracial state.