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

TL;DR

This paper provides a detailed structural analysis of the 2-adic ring C*-algebra $\

Contribution

It reveals the algebra's inner structure, automorphism groups, and maximal abelian subalgebras, offering new insights into its rigidity and symmetry properties.

Findings

The relative commutant $C^*(S_2)' \,\cap\, \mathcal{Q}_2$ is trivial.

The inclusion $\mathcal{O}_2 \subset \mathcal{Q}_2$ exhibits rigidity with no nontrivial endomorphisms fixing $\mathcal{O}_2$.

The automorphism group contains a maximal abelian subgroup topologically isomorphic to $C(\mathbb{T},\mathbb{T})$.

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 ${\rm Aut}(\mathcal{Q}_2)$ topologically isomorphic with $C(\mathbb{T},\mathbb{T})$. Finally, it is shown by an explicit construction that ${\rm Out}(\mathcal{Q}_2)$ is uncountable and non-abelian.