Ricerc@Sapienza

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems (G, P, θ). Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in 2 ≤ p < ∞ generators to the p-adic ring C∗algebra. For the case where P is abelian and C∗(G) is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix C∗(G) pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group.

We undertake a systematic study of the so-called 2-adic ring C‡-algebra Q2. This is the
universal C‡-algebra generated by a unitary U and an isometry S2 such that S2U = U2S2
and S2S‡
2+US2S‡
2U‡ = 1. Notably, it contains a copy of the Cuntz algebra O2 = C‡(S1;S2)
through the injective homomorphism mapping S1 to US2. Among the main results, the
relative commutant C‡(S2)œ 9 Q2 is shown to be trivial. This in turn leads to a rigidity
property enjoyed by the inclusion O2 ` Q2, namely the endomorphisms of Q2 that restrict