Wednesday, January 17, 2018 - 15:00
Speaker: Stefano Rossi, Università di Roma 2 Tor Vergata
Title: Endomorfismi e automorfismi della C*-algebra diadica
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_2^*U^*=1. By its very definition it contains a copy of the Cuntz algebra O_2. I'll start with an overview of some interesting properties of this inclusion, as they came to be pointed out in a joint work with V. Aiello and R. Conti. Among other things, the inclusion enjoys a kind of rigidity property, namely any endomorphism of the larger that restricts trivially to the smaller is trivial itself. I'll also say a word ot two about the extension problem, which asks whether an endomorphism of O_2 extends to an endomorphism of Q_2. To the best of our knowledge, an endomorphism of the former hardly ever extends to the latter. For instance, a good many examples of non-extendible endomorphisms show up as soon as the so-called Bogoljubov automorphisms of O_2 are looked at. I'll then move onto 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, entails a preliminary study of the inner structure of Q_2. More precisely, it's crucial to prove that C*(U) is a maximal commutative subalgebra. Time permitting, I'll also report on generalizations to considerably broader classes of C*-algebras dealt within a recently submitted paper with N. Stammeier as well.