Reference Request: Standard trace on $C^*(G)$

Question

Let $G$ be a discrete group and consider its group $C^*$-algebra, $C^*(G)$.

In some bibliography I've seen the term "let $tr$ be the standard trace on $C^*(G)$", but I'm not sure what it means, however from the context seems to be implied that it verifies the following propery.

Let $u:G \rightarrow C^*(G)$ be the universal representation, then $$ tr(u_g) = \begin{cases} 1 \quad &\text{if } g = e \\ 0 &\text{otherwise.} \end{cases} $$

My questions are as follows:

1. What is the standard trace on $C^*(G)$ and do you know a reference for it?
2. Does it verify the above property?

In the commutative case, one can use the Gelfand transform to show that $C^*(G) \cong C(\widehat G)$. Let $\phi_g$ be the image of $u_g$ under this isomorphism.

3. In the commutative setting, do $tr$ and this isomorphism induce a trace $\rho: C(\widehat G) \rightarrow \mathbb C$, such that for all $g \in G$,

$$ \rho(\phi_g)= \int_{\widehat G} \, \chi(g)d\mu(\chi), $$ where $\mu$ is the normalized Haar measure on $\widehat G$? If question 2) is true, then this means in particular that $$ \rho(\phi_g) = \begin{cases} 1 \quad &\text{if } g = e \\ 0 &\text{otherwise.} \end{cases} $$

PS: If given a reference I can probably work the rest out on my own, Thanks in advance!

Answer 1

Well, there is a standard trace on $C^*(G)$, but to describe it, we first describe the standard trace of $C^*_r(G)$.

Recall that the left regular representation of $G$ is the unitary representation $\lambda:G\to B(\ell^2G)$, given by $g\mapsto\lambda_g$, where $\lambda_g:\ell^2G\to\ell^2G$ is the unitary operator satisfying $\lambda_g(\xi_t)=\xi_{gt}$, where $\{\xi_t\}_{t\in G}$ is the canonical orthonormal basis of $\ell^2G$. The reduced group $C^*$-algebra of $G$ is by definition $C^*_r(G):=\overline{\text{span}}\{\lambda_g:g\in G\}\subset B(\ell^2(G))$.

Now $C^*_r(G)$ admits a standard trace, given by $\tau:C^*_r(G)\to\mathbb{C}$, $\tau(x)=\langle x\xi_{1_G},\xi_{1_G}\rangle_{\ell^2G}$. It is easy to check that $\tau$ is indeed a trace; Moreover, $\tau(\lambda_g)=\langle\xi_g,\xi_{1_G}\rangle=\delta_{g,1_G}$ (Kronecker delta).

Exercise: show that $\tau$ is faithful, i.e. $\tau(x^*x)=0\implies x=0$ for any $x\in C^*_r(G)$

(hint: there is a right regular representation of $G$ on $\ell^2G$ as well, and it has commuting range with the left regular rep).

By the universal property of $C^*(G)$, we have a canonical surjection $q:C^*(G)\to C^*_r(G)$ satisfying $q(u_g)=\lambda_g$ for all $g\in G$. Considering $\tau\circ q:C^*(G)\to\mathbb{C}$, we obtain a trace on $C^*(G)$ (check that it is a trace) mapping $u_g$ to $\delta_{g,1_G}$.

The reference I would give for this is Brown-Ozawa, 2.5. Check also Blackadar's encyclopedia on Operator algebras; otherwise Ken Davidson's "$C^*$-algebras by example", chapter VII is a good source.