TITLE: Topological K-theory for commutative C*-algebras
QUESTION [5 upvotes]: 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)?

REPLY [9 votes]: 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.