Separative Cancellation in Regular Rings and Exchange Rings
by
K. R. Goodearl
University of California at Santa Barbara

We shall discuss the development of two common threads tying together various open questions concerning cancellation in both (von Neumann) regular rings and an important class of C*-algebras, those having `real rank zero'. One connection is that the C*-algebras of real rank zero turn out to be precisely those C*-algebras which are exchange rings. (It has long been known that regular rings are exchange rings.) The parallels among cancellation problems for regular rings and C*-algebras of real rank zero thus simply reflect this fact. Many of these cancellation problems in turn are interconnected by the condition of separative cancellation for finitely generated projective modules:

A \oplus A =~ A \oplus B =~ B \oplus B     ===>     A =~ B.

It turns out that a number of open questions, including some that do not appear to involve cancellation at all, have positive answers for any exchange ring R satisfying separativity - for example,

\bullet If R is simple and directly finite, then finitely generated projective R-modules cancel from direct sums (equivalently, R has stable rank 1).

\bullet The stable rank of R is 1, 2, or \infty.

\bullet All regular square matrices over R are equivalent to diagonal matrices.

\bullet All invertible square matrices over R can be diagonalized by elementary row or column operations.

Although this harvest of positive results makes separativity appear quite strong, no non-separative exchange rings are known.

The results mentioned are from joint work with Pere Ara, Kevin O'Meara, Enric Pardo, and Robert Raphael.

Date received: December 21, 1998