A Look at the Inner Structure of the 2-adic Ring $C^*$‡-algebra and Its Automorphism Groups

Abstract

We undertake a systematic study of the so-called 2-adic ring $C^*$-algebra $\mathcal Q_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal O_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal Q_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal O_2\subset\mathcal Q_2$, namely the endomorphisms of $\mathcal Q_2$ that restrict to the identity on $\mathcal O_2$ are actually the identity on the whole $\mathcal Q_2$. Moreover, there is no conditional expectation from $\mathcal Q_2$ onto $\mathcal O_2$. As for the inner structure of $\mathcal Q_2$, the diagonal subalgebra $\mathcal D_2$ and $C^*(U)$ are both proved to be maximal abelian in $\mathcal Q_2$. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of $\mathcal Q_2$. In particular, the semigroup of the endomorphisms fixing $U$ turns out to be a maximal abelian subgroup of Aut$(\mathcal Q_2)$ topologically isomorphic with $C(\mathbb{T},\mathbb{T})$. Finally, it is shown by an explicit construction that Out$(\mathcal Q_2)$ is uncountable and non-abelian.