Dipartimento di Scienze di Base e Applicate per l'Ingegneria

Aula 1B1:

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.