Refinement Monoids with Weak Comparability and Applications to Regular Rings and C∗-algebras

We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O’Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and C∗-algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules. Introduction Let M be an (abelian) monoid. For x, y ∈M we will write x ≤ y if there exists z ∈M such that y = x+z. This preorder is sometimes called the algebraic preorder in M . Let M∗ denote the set of nonzero elements of M . M is called conical if M∗ is closed under addition. We will write x ≤∗ y if there exists z ∈ M∗ such that y = x + z. Note that the relation ≤∗ is transitive if M is a conical monoid. An order-unit in M is a nonzero element u ∈M such that for each x ∈M there exists n ≥ 1 such that x ≤ nu. A monoid M is said to be simple if it is nonzero and every nonzero element of M is an order-unit, so that M has no nontrivial ideals (i.e. convex submonoids of M). Definition. Let (M,u) be a monoid with order-unit. We say that (M,u) satisfies weak comparability provided that for all nonzero elements x in M such that x ≤ u, there exists a positive integer k = k(x) such that, if y ∈M and ky ≤ u, then y ≤ x. Note that, for conical M , if there is an element x′ ∈ M∗ such that x′ ≤∗ x, then replacing k(x) by k(x′) we obtain a positive integer k such that ky ≤ u implies y ≤∗ x. We say that a subset X of a monoid M is cancellative (respectively strictly cancellative) if, for a, b, c ∈ X , the relation a + c = b + c (resp. a + c ≤∗ b + c) implies a = b (resp. a ≤∗ b). Our main results are the following: Received by the editors May 19, 1994 and, in revised form, September 21, 1994. 1991 Mathematics Subject Classification. Primary 16E20, 16E50, 46L80, 19K14, 06F20.