Perforated Ordered K₀-Groups

Abstract

Abstract. A simple C * -algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition-induced by addition of orthogonal projections-form the additive semigroup {0, 2, 3, . . . }.(This is a particularly simple instance of the phenomenon of perforation of the ordered K 0 -group, which has long been known in the commutative case-for instance, in the case of the four-sphere-and was recently observed by the second author in the case of a simple C * -algebra.)

On the Independence of K-Theory and Stable Rank for Simple C*-Algebras

“…Section 2 lists several theorems from [3], which are applied in section 3 to construct the algebra B n of Theorem 1.1. The general ideas of this latter section are also found in [3]. In section 4 B n is shown to have the properties claimed in Theorem 1.1.…”

Embedding rank one simple groups into rank one simple Riesz groups

“…Recent results of Villadsen [14], R(rdam and Villadsen [13], and Elliott and Villadsen [4], guarantee the existence of simple C Ã -algebras whose K 0 groups are simple components. Hence, these results open the possibility of finding a simple C Ã -algebra of real rank zero A with stable rank one such that ðK 0 ðAÞ; K 0 ðAÞ þ Þ is isomorphic to ð e G GðA 2 ; B 9 ; H 2;7 ; mÞ; e G G þ ðA 2 ; B 9 ; H 2;7 ; mÞÞ for some odd infinite generalized integer m. Suppose that it is possible to construct such a -unital, nonunital, nonartinian, simple, stable rank one C Ã -algebra of real rank zero (von Neumann regular ring) A, and suppose that the interval D constructed in Lemma 3.1 is generated by f½e n g & VðAÞ, where fe n g is a -unit for A.…”

Monoids of Intervals of Simple Refinement Monoids and Non-stable K-Theory of Multiplier Algebras

“…The examples obtained in these works allow to construct monoids of intervals satisfying special pathologies, such as failure of separativity of the monoid of intervals (see [1]), among others. Also, Villadsen [25], Rørdam and Villadsen [21], Elliott and Villadsen [7], and Toms [22] constructed examples of simple C * -algebras of stable rank one whose K 0 groups are torsion free and strictly perforated. These examples suggests the possibility of constructing C * -algebras A with real rank zero, stable rank one, with K 0 (A) being strictly perforated.…”