Approximately finite-dimensional C*-algebra
The counterpart of simple AF C*-algebras in the von Neumann algebra world are the hyperfinite factors, which were classified by Connes and Haagerup.
An arbitrary finite-dimensional C*-algebra A takes the following form, up to isomorphism: where Mi denotes the full matrix algebra of i × i matrices.
Up to unitary equivalence, a unital *-homomorphism Φ : Mi → Mj is necessarily of the form where r·i = j.
In general, a unital homomorphism between finite-dimensional C*-algebras is specified, up to unitary equivalence, by a t × s matrix of partial multiplicities (rl k) satisfying, for all l In the non-unital case, the equality is replaced by ≤.
Consider the category whose objects are isomorphism classes of finite-dimensional C*-algebras and whose morphisms are *-homomorphisms modulo unitary equivalence.
By the above discussion, the objects can be viewed as vectors with entries in N and morphisms are the partial multiplicity matrices.
For instance, the Pascal triangle, with the nodes connected by appropriate downward arrows, is the Bratteli diagram of an AF algebra.
Given a Bratteli diagram of A and some subset S of nodes, the subdiagram generated by S gives inductive system that specifies an ideal of A.
It has its origins in topological K-theory and serves as the range of a kind of "dimension function."
Two elements p and q are said to be Murray-von Neumann equivalent, denoted by p ~ q, if p = vv* and q = v*v for some partial isometry v in M∞(A).
Define a binary operation + on the set of equivalences P(A)/~ by where ⊕ yields the orthogonal direct sum of two finite-dimensional matrices corresponding to p and q.
While we could choose matrices of arbitrarily large dimension to stand in for p and q, our result will be equivalent regardless.
K0(A) carries a natural order structure: we say [p] ≤ [q] if p is Murray-von Neumann equivalent to a subprojection of q.
Two preliminary facts are needed before one can sketch a proof of Elliott's theorem.
The proof of the lemma is based on the simple observation that K0(A) is finitely generated and, since K0 respects direct limits, K0(B) = ∪n βn* K0 (Bn).
Given an isomorphism between dimension groups, one constructs a diagram of commuting triangles between the direct systems of A and B by applying the second lemma.
We sketch the proof for the non-trivial part of the theorem, corresponding to the sequence of commutative diagrams on the right.
This specifies the range of the classifying functor K0 for AF-algebras and completes the classification.
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+.
It is clear that if A is finite-dimensional, (K0, K0+) is a Riesz group, where Zk is given entrywise order.
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.
Because each H1 has the canonical order structure, G1 is a direct sum of Z 's (with the number of copies possible less than that in H1).
Induction gives a directed system whose K0 is with scale This proves the special case.
In a related context, an approximately finite-dimensional, or hyperfinite, von Neumann algebra is one with a separable predual and contains a weakly dense AF C*-algebra.
Murray and von Neumann showed that, up to isomorphism, there exists a unique hyperfinite type II1 factor.
Powers exhibited a family of non-isomorphic type III hyperfinite factors with cardinality of the continuum.
