Embedding rank one simple groups into rank one simple Riesz groups

Abstract

Abstract. We give a method for embedding a large family of partially ordered simple groups of rank one into simple Riesz groups of rank one. In particular, we answer in the affirmative a question of Wehrung, by constructing a torsion-free, simple Riesz group G of rank one containing an intervalWe sketch some potential applications of this result in the context of monoids of intervals and K-Theory of rings.

Simple Riesz groups of rank one having wild intervals

“…Hence, these results suggest the problem of constructing simple C * -algebras A with real rank zero and stable rank one such that (K 0 (A), K 0 (A) + ) is isomorphic to one of the groups we construct in this paper (as well as those constructed in [14,15]), by lifting connecting maps in the direct limit expression of these groups (as limits of simple components and order-embeddings), to C * -algebra maps between C * -algebras of the type constructed in [24]. Other relevant aspects of this discussion are outlined in Section 7.…”

“…As mentioned in the introduction, one of the main objectives in [15] was to study the embedding of a certain class of simple partially ordered groups of rank one into simple Riesz groups of rank one. Such groups are parametrised by a triple (A, B, H), where H is a sequence of simple groups (basically Z with different positive cones) and A and B sequences of positive integers, all subject to certain axioms.…”

“…Such groups are parametrised by a triple (A, B, H), where H is a sequence of simple groups (basically Z with different positive cones) and A and B sequences of positive integers, all subject to certain axioms. The proof of the key embedding result, established in [15,Theorem 2.11], is based on the fact that these groups are isomorphic to a direct limit of an inductive system ((Z, G + n ), f n,n+1 ) such that, for every n in N, the map f n,n+1 :…”

“…Wehrung then asked whether such an example can be realised as a torsion-free Riesz group of rank one (i.e., with positive cone isomorphic to an additive submonoid of Q), see [31,Problem 3.15]. This question was answered by the second author in [15] by constructing a large family of simple groups that can be embedded into simple Riesz groups of rank one [15,Theorem 2.11].…”

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

“…to positive cones of simple Riesz groups), and to translate the results we obtain to the K-theoretical context. In this line is essential to consider into our scope recent results of Wehrung [32], and Pardo [15], [18], where some methods for constructing large families of strictly perforated simple Riesz groups are introduced. 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.…”