Abstract

The 2-adic ring C-algebra is the universal C-algebra Q_2 generated by an isometry S_2 and a unitary U such that S_2U=U^{2S_2 and S_2S_2}+US_2S_2U=1. By its very definition it contains a copy of the Cuntz algebra O_2. I'll start by discussing some nice properties of this inclusion, as they came to be pointed out in a recent joint work with V. Aiello and R. Conti. Among other things, the inclusion enjoys a kind of rigidity property, i.e., any endomorphism of the larger that restricts trivially to the smaller must be trivial itself. I'll also say a word or two about the extension problem, which is concerned with extending an endomorphism of O_2 to an endomorphism of Q_2. As a matter of fact, this is not always the case: a wealth of examples of non-extensible endomorphisms (automophisms indeed!) show up as soon as the so-called Bogoljubov automorphisms of O_2 are looked at. Then I'll move on to particular classes of endomorphisms and automorphisms of Q_2, including those fixing the diagonal D_2. Notably, the semigroup of the endomorphisms fixing U turns out to be a maximal abelian group isomorphic with the group of continuous functions from the one-dimensional torus to itself. Such an analysis, though, calls for some knowledge of the inner structure of Q_2.  More precisely, it's vital to prove that C(U) is a maximal commutative subalgebra. Time permitting, I'll also try to present forthcoming generalizations to broader classes of C*-algebras, on which we're currently working with N. Stammeier as well.