A look at the inner structure of the 2-adic ring C*-algebra and its automorphism groups

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)′ ∩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 to the identity on O2 are actually the identity on the whole Q2. Moreover, there is no conditional expectation from Q2 onto O2. As for the inner structure of Q2, the diagonal subalgebra D2 and C*(U)are both proved to be maximal abelian in Q2. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of Q2. In particular, the semigroup of the endomorphisms xing U turns out to be a maximal abelian subgroup of Aut(Q2)topologically isomorphic with C(T;T). Finally, it is shown by an explicit construction that Out(Q2) is uncountable and non- abelian.