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"?

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 $\tau(x) = \langle \delta_e, x \, \delta_e \rangle$.

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    $\begingroup$ Is the answer also for general locally compact groups? or just discrete? $\endgroup$ – Shirly Geffen Jan 27 at 15:07
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    $\begingroup$ Perhaps I should add a clarification to the answer. The characterization of amenability works in the case of locally compact groups but with a modification. In that case $\tau$ is not a trace but a tracial weight (in the case of unimodular groups) or a non-tracial weight in the case of general non-unimodular groups. In either case it still holds that the unbounded functional $\tau: C_\lambda^\ast(G) _+ \to [0,\infty]$ (now is only well-defined in the positive cone of the algebra) is a faithful weight iff $G$ is amenable. $\endgroup$ – Adrián González-Pérez Jan 27 at 15:13
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    $\begingroup$ For the first question: It also holds that you can extend $\tau$ to a weight -not a conditional expectation- (although some people think of weights as "unbounded conditional expectations"). Of course, it would not be given by a vector state but by a sum of vector states. In the case in which the group is second countable you can take that sum to be countable. The construction of this is a little bit more technical. It is described in detail in Pedersen's book "$C^\ast$-algebras and their automorphism groups". $\endgroup$ – Adrián González-Pérez Jan 27 at 15:17
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    $\begingroup$ For the first question, if you want $\tau$ to be a conditional expectation you need to impose that $G$ is discrete. Otherwise $\tau(\mathbb{1})$ would be infinity since, intuitively, $\lambda_e$ is given by $f$ being a Dirac delta over the unit $e \in G$. And evaluation in $e$ would give $\infty$. This argument can be made rigorous by taking an increasing approximate unit on $C_\lambda^\ast(G)$ $\endgroup$ – Adrián González-Pérez Jan 27 at 15:22

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