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

Abstract

We undertake a systematic study of the so-called 2-adic ring $C^{\ast }$-algebra ${\mathcal{Q}}_2$. This is the universal $C^{\ast }$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_{2}U=U^2{S}_2$ and $S_2{S}_2^{\ast }+U{S}_2{S}_2^{\ast }{U}^{\ast }=1$. Notably, it contains a copy of the Cuntz algebra ${\mathcal{O}}_2=C^{\ast }(S_1,S_2)$ through the injective homomorphism mapping $S_1$ to $U{S}_2$. Among the main results, the relative commutant $C^{\ast }(S_2{)}^{\prime }\cap {\mathcal{Q}}_2$ 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 $C^{\ast }(U)$ 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.