Interpretation of the $K$-groups of a $C^*$-algebra.

Question

I'm learning K-theory of $C^*$-algebras after studying algebraic topology and (co-)homology theories for topological spaces (especially singular and cellular homology, but I'm not familiar with topological K-theory in general). For topological spaces, there are interpretations what i.e. singular homology measures ($H_0$ measures the number of holes in the space and so forth).

Now I want to know how to interpret $K_0(A)$ and $K_1(A)$ of a $C^*$-algebra $A$.

Definitions. For $K_0$ I know the following definition:

1. For a unital $C^*$-algebra A: Let $P_{\infty}(A)=\bigcup\limits_{n\in\mathbb{N}} P_n(A)$, where $P_n(A)=\{p\in M_n(A): p^2=p^*=p\}$ (and $P_n\hookrightarrow P_{n+1}$ for all n in the canonical way), endowed with $$\bigoplus :P_{\infty}(A)\times P_{\infty}(A)\to P_{\infty}(A),\; (p,q)\mapsto diag_{m+n}(p,q)$$ for $p\in P_m(A), q\in P_n(A)$. Endow $P_{\infty}(A)$ with an equivalence relation: For $p\in P_m(A), q\in P_n(A)$ let $p \sim q:\iff $ there exists a $v\in M_{n,m}(A)$ such that $p=v^*v$ and $q=vv^*$. Set $D(A):=P_{\infty}(A)/\sim$ and define $K_0(A)$ as the Grothendieck group of the abelian monoid $(D(A),\bigoplus)$.

   2. For $A$ nonunital: We have a split exact sequence $$0\to A \xrightarrow{i} A^+\xrightarrow{\pi} \mathbb{C}\to0$$ where $A^+$ is the one-point unitization of $A$, $i$ is the canonical embedding and $\pi(a+\lambda 1)=\lambda$. It is $K_0(A):=\ker (K_0(\pi))$.

For $K_1$ I know the following definition:

$K_1(A):=U_{\infty}(A^+)/\sim $, where: $U_{\infty}(A^+)=\bigcup\limits_{n\in\mathbb{N}} U_n(A^+)$, $U_n(A^+)=\{u\in M_n(A^+): u^*u=uu^*=1\}$ and for $u\in U_m(A^+), v\in U_n(A^+)$ let $u \sim v:\iff $ there exists a $k\ge max\{m,n\}$ such that $u\oplus 1_{k-m}$ is homotopic to $v\oplus 1_{k-n}$. $\oplus$ is defined analogously as above: $$\bigoplus :U_{\infty}(A^+)\times U_{\infty}(A^+)\to U_{\infty}(A^+),\; (u,v)\mapsto diag_{m+n}(u,v)$$ for $u\in U_m(A^+), v\in U_n(A^+)$.

Ideas/problems -I guess that $K_1(A)$ counts the number of path connected components of the unitaries $U_{\infty} (A^+)$, since a homotopy between such elements is a continuous path connecting them. Is it correct?

-I'm stuck to give a suitable interpretation for $K_0(A)$, if that is possible. I would appreciate your ideas here.

Regards.

Edit: I have found this very similar question now and there is a partial answer What properties of a C*-algebra are reflected in their $K_0$ groups?

Answer 1

Maybe this answers your question: We know that the $K$-groups are abelian. And in fact, every pair $(G_0,G_1)$ of countable abelian groups occurs as $(K_0(A),K_1(A))$ for some $C^*$-algebra $A$. Furthermore, the $K$-groups of a separable $C^*$-algebra are neccesarily countable.