Presented by:

Stefano Rossi Università degli Studi di Roma Tor Vergata, 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.

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