TITLE: $*$-algebras, completions, and $K$-theory
QUESTION [7 upvotes]: What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $A_1$ and $A_2$, such that the two associated $K$-theory groups are non-isomorphic, that is:
$$
K(A_1) \not\simeq K(A_2). 
$$

REPLY [11 votes]: Any infinite discrete group $\Gamma$ with Kazhdan's property (T) gives an example. Since it is not amenable, the full and reduced C*-algebras (which are both completions of the group algebra) do not coincide. Moreover, the full C*-algebra contains a projection with non-trivial K-theory class (so-called Kazhdan projection) which is mapped to $0$ in the reduced C*-algebra.
As for free groups, they are $K$-amenable, meaning that the canonical surjection between the full and reduced C*-algebras induces an isomorphism in $K$-theory.
The basic reference for this is J. Cuntz's article K-theoretic amenability for discrete groups, J. reine angew. Math. 1983 (where he also acknowledges overlapping work of E.C. Lance).