Purely Infinite C*-Algebras of Real Rank Zero

Cornel Pasnicu, University of Puerto Rico

Abstract. A C*-algebra can be thought of as a “non-commutative topological space.” Kirchberg and Rørdam introduced an important class of algebras: the purely infinite C*-algebras, motivated by a deep classification result of C*-algebras, obtained by Kirchberg, Elliott’s classification conjecture has been verified for a wide class of C*-algebras of real rank zero (“zero dimensional non-commutative topological spaces”). We will present a joint work with Mikael Rørdam, to appear in J. Reine Angew. Math., in which we characterize the real rank zero condition for a separable, purely infinite C*-algebra by a K-theoretical condition together with a topological condition for the primitive spectrum of the C*-algebra. Some interesting consequences of this result, including a pathology in the ideal structure of the (minimal) tensor product of two (non-exact) C*-algebras, will be given.