What is an example of a $*$algebra $\cal{A}$, which admits two nonequivalent 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 nonisomorphic, that is: $$ K(A_1) \not\simeq K(A_2). $$

2$\begingroup$ How about $C(\mathbf{F}_2)$ and $C^r(\mathbf{F}_2)$ which are completions of $\ell_1(\mathbf{F}_2)$ under suitable norms? $\endgroup$ – Tomek Kania Nov 21 '18 at 19:12

4$\begingroup$ @TomekKania Hi Tomek, I think that the canonical map from the full algebra of ${\bf F}_2$ onto the reduced algebra actually induces an isomorphism on Ktheory; if my memory is correct this is an Acta paper of E. C. Lance, but I haven't had time to check. (The relevant phrase is "Kamenability") $\endgroup$ – Yemon Choi Nov 21 '18 at 19:25
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 nontrivial Ktheory class (socalled 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 Ktheoretic amenability for discrete groups, J. reine angew. Math. 1983 (where he also acknowledges overlapping work of E.C. Lance).

1$\begingroup$ Nice answer  I suppose it is still possible that the $K$ groups are isomorphic, just not naturally ... $\endgroup$ – Nik Weaver Nov 21 '18 at 21:24