  We study the 2-adic version of the ring $C^*$-algebra of the integers. First,
we work out the precise relation between the Cuntz algebra $\cO_2$ and our
2-adic ring $C^*$-algebra in terms of representations. Secondly, we prove a
2-adic duality theorem identifying the crossed product arising from 2-adic
affine transformations on the 2-adic numbers with the analogous crossed product
algebra over the real numbers. And finally, as an outcome of this duality
result, we construct an explicit imprimitivity bimodule and prove that it
transports one canonical representation into the other.
