What is the unitary dual of the non discrete abelian group containing elements of the type $$ \left( \frac{k}{2^{j}},\frac{l}{2^{j}}\right) $$ where $k,l,j$ are integers? The group law here is regular addition. That is $$ \left( \frac{k}{2^{j}},\frac{l}{2^{j}}\right) +\left( \frac{k_{1}}{2^{s}},\frac{l_{1}}{2^{s}}\right) =\left( \frac{2^{s}k+2^{j}k_{1}}{2^{j+s}}% ,\frac{2^{s}l+2^{j}l_{1}}{2^{j+s}}\right) $$
Any input will be appreciated.
