DIAGONAL AUTOMORPHISMS OF THE 2-ADIC RING C*-ALGEBRA

The 2-adic ring C*-algebra Q(2) naturally contains a copy of the Cuntz algebra O-2 and, a fortiori, also of its diagonal subalgebra D-2 with Cantor spectrum. This paper is aimed at studying the group Aut(D2)(Q) of the automorphisms of Q(2) fixing D-2 pointwise. It turns out that any such automorphism leaves O-2 globally invariant. Furthermore, the subgroup Aut(D2)(Q(2)) is shown to be maximal abelian in Aut(Q(2)). Saying exactly what the group is amounts to understanding when an automorphism of O-2 that fixes D-2 pointwise extends to Q(2). 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.