A look at the inner structure of the 2-adic ring \(C^*\)-algebra and its automorphism groups (Q1707956)

Summary: 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 \(C^*(S_2)^\prime\cap \mathcal Q_2\) 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 \(C^*(U)\) 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 \(\mathrm{Aut}(\mathcal Q_2)\) topologically isomorphic with \(C(\mathbb{T},\mathbb{T})\). Finally, it is shown by an explicit construction that \(\mathrm{Out}(\mathcal Q_2)\) is uncountable and non-abelian.