# Endomorphisms and automorphisms of the 2-adic ring C*-algebra Q_2

Presented by:
Stefano Rossi Università degli Studi di Roma Tor Vergata
Date:
Thursday 16th February 2017 - 14:00 to 15:00
Venue:
INI Seminar Room 2
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_2^*U^*=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.

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.