page_title: Approximately finite-dimensional C*-algebra
section_title: Riesz groups
hierarchical_section_title: Approximately finite-dimensional C*-algebra / Classification / The Effros-Handelman-Shen theorem / Riesz groups
caption_attribution_description: Diagram for Shen's criterion
context_page_description: In mathematics, an approximately finite-dimensional C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the K₀ functor whose range consists of ordered abelian groups with sufficiently nice order structure.
  The classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple nuclear stably finite C*-algebras. Its proof divides into two parts. The invariant here is K₀ with its natural order structure; this is a functor. First, one proves existence: a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows uniqueness: the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as the intertwining argument. For unital AF algebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative.
context_section_description: A group G with a partial order is called an ordered group. The set G⁺ of elements ≥ 0 is called the positive cone of G. One says that G is unperforated if k·g ∈ G⁺  implies g ∈ G⁺.
  The following property is called the Riesz decomposition property: if x, yᵢ ≥ 0 and x ≤ ∑ yᵢ, then there exists xᵢ ≥ 0 such that x = ∑ xᵢ, and xᵢ ≤ yᵢ for each i.
  A Riesz group (G, G⁺) is an ordered group that is unperforated and has the Riesz decomposition property.
  It is clear that if A is finite-dimensional, (K₀, K₀⁺) is a Riesz group, where Zᵏ is given entrywise order. The two properties of Riesz groups are preserved by direct limits, assuming the order structure on the direct limit comes from those in the inductive system. So (K₀, K₀⁺) is a Riesz group for an AF algebra A.
  A key step towards the Effros-Handelman-Shen theorem is the fact that every Riesz group is the direct limit of Zᵏ 's, each with the canonical order structure. This hinges on the following technical lemma, sometimes referred to as the Shen criterion in the literature.
  Lemma Let (G, G⁺) be a Riesz group, ϕ: (Zᵏ, Zᵏ₊) → (G, G⁺) be a positive homomorphism. Then there exists maps σ and ψ, as indicated in the adjacent diagram, such that ker(σ) = ker(ϕ).
  Corollary Every Riesz group (G, G⁺) can be expressed as a direct limit
  where all the connecting homomorphisms in the directed system on the right hand side are positive.
caption: The Shen criterion.
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