04/03/2011, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
N. Christopher Phillips, University of Oregon, USA
Equivariant semiprojectivity and applications
A C*-algebra \(A\) is semiprojective if, roughly speaking, any
approximate homomorphism from \(A\) to some other C*-algebra is
close to an actual homomorphism. (There are several different ways
to make this precise, leading to several different concepts.) There
are not very many semiprojective C*-algebras, but the ones that do
exist (and the fact that they are semiprojective) play an important
role in the theory. Examples of semiprojective C*-algebras include
finite dimensional C*-algebras, the algebra of continuous functions
from the interval or circle to a finite dimensional C*-algebra, the
Cuntz algebras and some of their generalizations, and the full
C*-algebras of free groups. Now suppose that a compact group \(G\)
acts on \(A\). We say that \(A\) is equivariantly semiprojective
if, roughly speaking, whenever \(B\) is another C*-algebra with an
action of \(G\), then every approximately equivariant approximate
homomorphism from \(A\) to \(B\) is close to an exactly equivariant
homomorphism. We prove that finite dimensional C*-algebras are
equivariantly semiprojective, as well as Cuntz algebras with
certain special actions. One of the many applications of
semiprojectivity is to classification theorems. In the
classification of purely infinite simple C*-algebras,
semiprojectivity is used to replace asymptotic morphisms with
homomorphisms. We expect equivariant semiprojectivity to play the
same role in the classification of actions of compact groups on
purely infinite simple C*-algebras.