The 2-adic ring $C^*$-algebra of the integers and its representations

Details The 2-adic ring $C^*$-algebra of the integers and its representations The 2-adic ring $C^*$-algebra of the integers and its representations

2010-11-25

arxiv.org/abs/1011.5622v2

Mathematics

Operator Algebras

Changes to Remark 3.2. New Remark 6.5 and added paragraph after Proposition 6.3. Corrected typos

Scientific paper

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.

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