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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.

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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 $m$.

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    $\begingroup$ Thank you for your answer sir, but I don't understand the first paragraph: The group is infinite symmetric group, how do you know that? And second what is its $C^{*}$-algebra? Sorry if my questions are so dumb. $\endgroup$
    – Peg Leg Jonathan
    Commented Aug 29, 2020 at 16:46
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    $\begingroup$ @PegLegScott please don't take this the wrong way, but it really doesn't look like you have enough background to be studying K-theory of C*-algebras. Why not start a little further back? $\endgroup$
    – Nik Weaver
    Commented Aug 29, 2020 at 18:04
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    $\begingroup$ You wrote you consider the group of finite support bijections of $\mathbb{N}$. Such a bijection is the identity outside of a finite set and permutes the elements in that finite set. Therefore you can identify the group of finite support bijections with the union over all groups $S_n$. Does that help? $\endgroup$
    – Ulrich Pennig
    Commented Aug 29, 2020 at 18:47
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    $\begingroup$ As for the group $C^*$-algebra, I guess the answer is: It is what it is. You can view it as the direct limit over all $C^*$-algebras $C^*(S_n)$, which reveals it as an AF-algebra, if that helps. The algebras $C^*(S_n)$ are of course finite-dimensional and can be seen as direct sums of matrix algebras. $\endgroup$
    – Ulrich Pennig
    Commented Aug 29, 2020 at 18:49
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    $\begingroup$ Karen Strung has good notes about C*-algebras in the context of the classification programme. They also treat K-theory and can be found here: strung.me/karen/CStarIntroDraft.pdf . Maybe these are useful. $\endgroup$
    – Ulrich Pennig
    Commented Aug 29, 2020 at 19:11

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