Extreme points of the set of all traces Let $G$ be a finitely generated group with a bound on its complex unitary irreducible representations: That is assume all complex unitary irreducibles of $G$ have degrees at most $k$ for some integer $k$. For example all finite groups and all virtually abelian groups satisfy this constraint. See for example my other question Irreducibles of virtually abelian finitely generated groups.
Then by virtue of these finite dimensional irreducibles, we can define traces on the group algebra $\mathbb{C}[G]$: For any irreducible $\rho$ of $G$ we see that $a \mapsto \mathrm{tr}(\rho(a))$ is a trace on $\mathbb{C}[G]$.
By trace I mean any positive linear functional $\tau: \mathbb{C}[G] \to \mathbb{C}$ such that $\tau(ab) = \tau(ba)$ and $\tau(1) = 1$. Let $T$ be the set of all traces on $\mathbb{C}[G]$. We can easily see that $T$ is convex.
If $G$ is finite it is elementary to see that $T$ is a simplex and its finitely many extreme points are exactly the traces coming from the irreducible representations (these are essentially the irreducible characters).
Now if $G$ is not finite can we still say that all extreme points of $T$ are the traces coming from irreducibles of $G$?
Some notes:

*

*For these special groups $G$, there are many references studying the algebraic properties of $\mathbb{C}[G]$. For example we know this algebra is semiprimitive (i.e., its Jackobson radical is trivial) and semiprime. Note that when $G$ is not finite, $\mathbb{C}[G]$ is not Artinian (hence not semisimple). See the book by Donald Passman "Infinite group rings".


*Any reference that studies properties of group von Neumann algebras or group $C^*$-algebras of these representation bounded groups would be very helpful.
 A: Please read this as a long comment for your reference request about the group $C^{\ast}$-algebra and the group von-Neumann algebra.
Let $G$ be a locally compact group. It is common to denote
the group $C^{\ast}$-algebra by $C^{\ast}(G)$,
the reduced group $C^{\ast}$-algebra by $C_r^{\ast}(G)$,
the Fourier algebra by $A(G)$,
the Fourier-Stieltjes algebra by $B(G)$,
the group von Neumann algebra by $VN(G)$, and
the enveloping group von Neumann algebra by $W^{\ast}(G)$.
It is well known that $W^{\ast}(G) = (B(G))^{\ast}$, $B(G) = (C^{\ast}(G))^{\ast}$, $VN(G) = (A(G))^{\ast}$ as Banach spaces, e.g., please see the book by Kaniuth and Lau https://bookstore.ams.org/surv-231 .
Next, a von Neumann algebra $M$ is type I finite if and only if its predual $M_{\ast}$ has the Dunford-Pettis property, pls. see Lemma 1 in Hamana1977 and Proposition 5.1 in OikhbergSpinu2015 .
Also, the following are equivalent for any $C^{\ast}$-algebra $A$.

*

*$i.$ $A^{\ast\ast}$ is a finite type I von Neumann algebra.

*$i'.$ $A^{\ast\ast}$ is a finite von Neumann algebra.

*$ii.$ $A^{\ast}$ has the Dunford-Pettis property.

*$iii.$ $A$ has no infinite dimensional irreducible representations.

*$iv.$ $A$ has the Dunford-Pettis property.

$(i\Leftrightarrow i'\Leftrightarrow ii\Leftrightarrow iii)$ Theorem 1 in Hamana1977 and Bunce1992.
$(iii\Leftrightarrow iv)$ see Corollary 5.4 in OikhbergSpinu2015.

We can simply take $M=VN(G)$ and $A=C^{\ast}(G)$, $A=C^{\ast}_r(G)$ above. For instance, the following theorem is from LauÜlger1993.

If we also assume that every unitary representation of $G$ is atomic, or the regular representation is atomic, then we can say more, please see Taylor1983 and LauÜlger1993.
