Functional Analysis and Applications Seminar 

04/03/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
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.