I'm new to K-Theory for $C^{\*}$-algebra and $C^{\*}$-algebra of groups.


If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{\*}(X)$?


I was planning to calculate $C^{\*}(X)$ first and then its projections and their homotopy classes but I failed to determine $C^{\*}(X)$ in the first place.


Here is the definition of the $C^{\*}$-algebra of a group:


<https://pages.uoregon.edu/ncp/Courses/2016ShanghaiCrPrdFiniteGps/Slides/Lecture1_Print_NoP.pdf>


Can anybody help me?


Thanks a lot.



K-Theory of $C^{*}(X)$
The group you describe should be the infinite symmetric group $S\_{\infty}$. The $K$-theory of its $C^\*$-algebra has been determined by Kerov and Vershik in


The K -functor (Grothendieck group) of the infinite symmetric group
<https://link.springer.com/article/10.1007/BF02104985>


The main result can be summarised as follows: Let $\mathcal{A}$ be the ring of symmetric polynomials in infinitely many variables. This is isomorphic to $\mathbb{Z}[a\_1, a\_2, \dots]$, where $a\_i$ is the $i$th elementary symmetric function in an infinite number of arguments. Then
$$
K\_0(C^\*S\_{\infty}) \cong \mathcal{A}\,/\, (a\_1 - 1)\mathcal{A}\ .
$$
The isomorphism sends the irreducible representation $\pi\_{\lambda}$ corresponding to the Young diagram $\lambda$ to the Schur function corresponding to $\lambda$. This is also an isomorphism of rings, where the ring structure on the left hand side comes from the observation that
$$
K\_0(C^\*S\_{\infty}) \cong \lim\_n K\_0(C^\*S\_n)
$$
and multiplies two representations $\pi\_1 \colon S\_n \to GL(V)$ and $\pi\_1 \colon S\_m \to GL(W)$ to
$$
Ind\_{S\_n \times S\_m}^{S\_{n+m}}(\pi\_1 \otimes \pi\_2)\ ,
$$
where $S\_n \times S\_m$ sits inside $S\_{n+m}$ with $S\_n$ permuting the first $n$ elements and $S\_m$ permuting the other $m$.


