# A Look at the Inner Structure of the 2-adic Ring $C_{∗}$-algebra and Its Automorphism Groups

### Valeriano Aiello

Vanderbilt University, Nashville, USA### Roberto Conti

Università di Roma La Sapienza, Italy### Stefano Rossi

Università di Roma Tor Vergata, Italy

## Abstract

We undertake a systematic study of the so-called 2-adic ring $C_{∗}$-algebra $Q_{2}$. This is the universal $C_{∗}$-algebra generated by a unitary $U$ and an isometry $S_{2}$ such that $S_{2}U=U_{2}S_{2}$ and $S_{2}S_{2}+US_{2}S_{2}U_{∗}=1$. Notably, it contains a copy of the Cuntz algebra $O_{2}=C_{∗}(S_{1},S_{2})$ through the injective homomorphism mapping $S_{1}$ to $US_{2}$. Among the main results, the relative commutant $C_{∗}(S_{2})_{′}∩Q_{2}$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $O_{2}⊂Q_{2}$, namely the endomorphisms of $Q_{2}$ that restrict to the identity on $O_{2}$ are actually the identity on the whole $Q_{2}$. Moreover, there is no conditional expectation from $Q_{2}$ onto $O_{2}$. As for the inner structure of $Q_{2}$, the diagonal subalgebra $D_{2}$ and $C_{∗}(U)$ are both proved to be maximal abelian in $Q_{2}$. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of $Q_{2}$. In particular, the semigroup of the endomorphisms fixing $U$ turns out to be a maximal abelian subgroup of Aut$(Q_{2})$ topologically isomorphic with $C(T,T)$. Finally, it is shown by an explicit construction that Out$(Q_{2})$ is uncountable and non-abelian.

## Cite this article

Valeriano Aiello, Roberto Conti, Stefano Rossi, A Look at the Inner Structure of the 2-adic Ring $C_{∗}$-algebra and Its Automorphism Groups. Publ. Res. Inst. Math. Sci. 54 (2018), no. 1, pp. 45–87

DOI 10.4171/PRIMS/54-1-2