# Topological K-theory for commutative C*-algebras

It is in some sense folklore that given two arbitrary abelian groups $G,H$ one can find a $C^*$ algebra $A$ such that $K_0(A)=G$ and $K_1(A)=H$. My question is the following: what is known in the case of commutative $C*$-algebra? Which groups can be obtained as $K$ groups of commutative $C^*$ algebra (in other words, $K$ groups for topological spaces)?

• In "Introduction to K-theory for C*-algebras" by Rordam, et.al. in the remark after Exercise 13.2 they mention that all pairs of countable abelian groups can arise as $K_*$ for some abelian C*-algebra. They explicitly point out how to do it in the finitely generated case. My K-theory knowledge is too weak to be of any more use. Commented Jun 7, 2014 at 1:49
• @truebaran I think that a finite group can Not be isomorphic to $K^{0}(X)$ for a topological space, since $K^{0}(X)$ must contain $\mathbb{Z}$ as a summand(subgroup). So there are some restriction. So a question: Assume that $\mathbb{Z}$ is a summand of an abelian group $G$. Is $G$ isomorphic to $K^{0}(X)$ for some $X$? Commented Nov 30, 2014 at 18:50
• @truebaran Are you considering complex K theory or real K theory? Commented Nov 30, 2014 at 19:04
• Unless you're specifically dealing with real $C^\ast$-algebras, everything in sight will be over the complex numbers. Commented Dec 1, 2014 at 10:00

Any pair of abelian groups arises, up to isomorphism, as the $K$-theory groups of a commutative $C^*$-algebra. If the groups were countable, the $C^*$-algebra can be chosen to be separable. This follows from Corollary 23.10.3 in Blackadar's $K$-theory book.
For a CW complex X we have $K_0(C(X)) \otimes \mathbb{Q} \cong \bigoplus_n H_{2n}(X; \mathbb{Q})$ and $K_1(C(X)) \otimes \mathbb{Q} \cong \bigoplus_n H_{2n+1}(X; \mathbb{Q})$.
The torsion subgroups may be harder, and the order structure on $K_0$ is notoriously tricky for $C(X)$.