Diagonal automorphisms of the 2-adic ring C∗-algebra

The 2-adic ring C∗-algebra Q2naturally contains a copy of the Cuntz algebra O2 and, a fortiori, also of its diagonal subalgebra D2 with Cantor spectrum. This paper is aimed at studying the group AutD2(Q2) of the automorphisms of Q2fixing D2 pointwise. It turns out that any such automorphism leaves O2 globally invariant. Furthermore, the subgroup AutD2(Q2) is shown to be maximal abelian in Aut(Q2). Saying exactly what the group is amounts to understanding when an automorphism of O2 that fixes D2 pointwise extends to Q2. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.