Condition expectation on maximal group C*-algebra

Question

I am trying to understand some basic $C^*$-algebraic terminology in a context specific to group $C^*$-algebras, and would appreciate some help from experts to whom the meaning of these things is clear.

Suppose $G$ is a locally compact topological group, and let $C^*(G)$ denote the maximal group $C^*$-algebra. Consider the map

$$C_c(G)\rightarrow\mathbb{C},\qquad f\mapsto f(e).$$

I am wondering:

1. Does this map extend to what is called a "conditional expectation" on $C^*(G)$?
2. If so, is this conditional expectation "faithful"?

Answer 1

The answer to the second is negative. In fact, it is a well known characterization of amenability.

Indeed, a group $G$ is amenable iff the trace $\tau:C^\ast(G) \to \mathbb C$, that you define as $$f \mapsto f(e)$$ is faithful. To see that, just use that if $\tau$ is faithful so shall its associated GNS representation be. But the image of the associated GNS is $C_\lambda^\ast(G)$ and $C_\lambda^\ast(G)$ and $C^\ast(G)$ are isomorphic iff $G$ is amenable.

The first is true since $\tau$ is given by composing $q:C^\ast(G) \to C^\ast_\lambda(G)$ with the trace of $C_\lambda^\ast(G)$. But you can express that trace as a vector state $\tau(x) = \langle \delta_e, x \, \delta_e \rangle$.