Ricerc@Sapienza

We undertake a systematic study of the so-called 2-adic ring C‡-algebra Q2. This is the
universal C‡-algebra generated by a unitary U and an isometry S2 such that S2U = U2S2
and S2S‡
2+US2S‡
2U‡ = 1. Notably, it contains a copy of the Cuntz algebra O2 = C‡(S1;S2)
through the injective homomorphism mapping S1 to US2. Among the main results, the
relative commutant C‡(S2)œ 9 Q2 is shown to be trivial. This in turn leads to a rigidity
property enjoyed by the inclusion O2 ` Q2, namely the endomorphisms of Q2 that restrict

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.