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:
$$
K(A\_1) \not\simeq K(A\_2). 
$$



$*$-algebras, completions, and $K$-theory
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](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0344?tify=%7B%22pages%22:[184],%22view%22:%22info%22%7D), J. reine angew. Math. 1983 (where he also acknowledges overlapping work of E.C. Lance). 


