Summary: K 0 OF PURELY INFINITE SIMPLE REGULAR RINGS
P. ARA, K. R. GOODEARL, AND E. PARDO
Abstract. We extend the notion of a purely infinite simple C*­algebra to the context
of unital rings, and we study its basic properties, specially those related to K­Theory.
For instance, if R is a purely infinite simple ring, then K 0 (R) + = K 0 (R), the monoid
of isomorphism classes of finitely generated projective R­modules is isomorphic to the
monoid obtained from K 0 (R) by adjoining a new zero element, and K 1 (R) is the
abelianization of the group of units of R. We develop techniques of construction,
obtaining new examples in this class in the case of von Neumann regular rings, and we
compute the Grothendieck groups of these examples. In particular, we prove that every
countable abelian group is isomorphic to K 0 of some purely infinite simple regular ring.
Finally, some known examples are analyzed within this framework.
Introduction
In 1981, Cuntz [12] introduced the concept of a purely infinite simple C*­algebra.
This notion has played a central role in the development of the theory of C*­algebras
in the last two decades. A large series of contributions, due to Blackadar, Brown,
Lin, Pedersen, Phillips, Rørdam and Zhang, among others, reflect the interest in the
structure of such algebras. One of the most important advances in the program of
classifying separable C*­algebras through K­Theory, proposed by Elliott in the early
seventies, was obtained in this context by Kirchberg [16] and Phillips [25], who showed

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics