Permutative representations of the 2-adic ring C*-algebra

The notion of permutative representation is generalized to the 2- adic ring C*-algebra Q2. Permutative representations of Q2 are then investigated with a particular focus on the inclusion of the Cuntz algebra O2 ⊂ Q2. Notably, every permutative representation of O2 is shown to extend automatically to a permutative representation of Q2 provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of O2 are proved to be unitarily equivalent to one another. Irreducible permutative representations of Q2 are classified in terms of irreducible permutative representations of the Cuntz algebra. Apart from the canonical representation of Q2, every irreducible representation of Q2 is the unique extension of an irreducible permutative representation of O2. Furthermore, a permutative representation of Q2 will decompose into a direct sum of irreducible permutative subrepresentations if and only if it restricts to O2 as a regular representation in the sense of Bratteli-Jorgensen. As a result, a vast class of pure states of O2 is shown to enjoy the unique pure extension property with respect to the inclusion O2 ⊂ Q2.