Finite-dimensional C*-algebras

An arbitrary finite-dimensional C*-algebra A takes the following form, up to isomorphism:

${\displaystyle \oplus _{k}M_{n_{k}},}$

where M_i denotes the full matrix algebra of i × i matrices.

Up to unitary equivalence, a unital *-homomorphism Φ : M_i → M_j is necessarily of the form

${\displaystyle \Phi (a)=a\otimes I_{r},}$

where r·i = j. The number r is said to be the multiplicity of Φ. In general, a unital homomorphism between finite-dimensional C*-algebras

${\displaystyle \Phi$ :\oplus _{1}^{s}M_{n_{k}}\rightarrow \oplus _{1}^{t}M_{m_{l}}}

is specified, up to unitary equivalence, by a t × s matrix of partial multiplicities (r_{l k}) satisfying, for all l

${\displaystyle \sum _{k}r_{lk}n_{k}=m_{l}.\;}$

In the non-unital case, the equality is replaced by ≤. Graphically, Φ, equivalently (r_{l k}), can be represented by its Bratteli diagram. The Bratteli diagram is a directed graph with nodes corresponding to each n_k and m_l and the number of arrows from n_k to m_l is the partial multiplicity r_lk.

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.

AF algebras

A C*-algebra is AF if it is the direct limit of a sequence of finite-dimensional C*-algebras:

${\displaystyle A=\varinjlim \cdots \rightarrow A_{i}\,{\stackrel {\alpha _{i}}{\rightarrow }}A_{i+1}\rightarrow \cdots ,}$

where each A_i is a finite-dimensional C*-algebra and the connecting maps α_i are *-homomorphisms. We will assume that each α_i is unital. The inductive system specifying an AF algebra is not unique. One can always drop to a subsequence. Suppressing the connecting maps, A can also be written as

${\displaystyle A={\overline {\cup _{n}A_{n}}}.}$

The Bratteli diagram of A is formed by the Bratteli diagrams of {α_i} in the obvious way. For instance, the Pascal triangle, with the nodes connected by appropriate downward arrows, is the Bratteli diagram of an AF algebra. A Bratteli diagram of the CAR algebra is given on the right. The two arrows between nodes means each connecting map is an embedding of multiplicity 2.

${\displaystyle 1\rightrightarrows 2\rightrightarrows 4\rightrightarrows 8\rightrightarrows \dots }$

(A Bratteli diagram of the CAR algebra)

If an AF algebra A = (∪_nA_n)⁻, then an ideal J in A takes the form ∪_n (J ∩ A_n)⁻. In particular, J is itself 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. In fact, every ideal arises in this way.

Due to the presence of matrix units in the inductive sequence, AF algebras have the following local characterization: a C*-algebra A is AF if and only if A is separable and any finite subset of A is "almost contained" in some finite-dimensional C*-subalgebra.

The projections in ∪_nA_n in fact form an approximate unit of A.

It is clear that the extension of a finite-dimensional C*-algebra by another finite-dimensional C*-algebra is again finite-dimensional. More generally, the extension of an AF algebra by another AF algebra is again AF.^[1]

K₀

The K-theoretic group K₀ is an invariant of C*-algebras. It has its origins in topological K-theory and serves as the range of a kind of "dimension function." For an AF algebra A, K₀(A) can be defined as follows. Let M_n(A) be the C*-algebra of n × n matrices whose entries are elements of A. M_n(A) can be embedded into M_{n + 1}(A) canonically, into the "upper left corner". Consider the algebraic direct limit

${\displaystyle M_{\infty }(A)=\varinjlim \cdots \rightarrow M_{n}(A)\rightarrow M_{n+1}(A)\rightarrow \cdots .}$

Denote the projections (self-adjoint idempotents) in this algebra by P(A). 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). It is clear that ~ is an equivalence relation. Define a binary operation + on the set of equivalences P(A)/~ by

${\displaystyle [p]+[q]=[p\oplus q]}$

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. This makes P(A)/~ a semigroup that has the cancellation property. We denote this semigroup by K₀(A)⁺. Performing the Grothendieck group construction gives an abelian group, which is K₀(A).

K₀(A) carries a natural order structure: we say [p] ≤ [q] if p is Murray-von Neumann equivalent to a subprojection of q. This makes K₀(A) an ordered group whose positive cone is K₀(A)⁺.

For example, for a finite-dimensional C*-algebra

${\displaystyle A=\oplus _{k=1}^{m}M_{n_{k}},}$

one has

${\displaystyle (K_{0}(A),K_{0}(A)^{+})=(\mathbb {Z} ^{m},\mathbb {Z} _{+}^{m}).}$

Two essential features of the mapping A ↦ K₀(A) are:

1. K₀ is a (covariant) functor. A *-homomorphism α : A → B between AF algebras induces a group homomorphism α_* : K₀(A) → K₀(B). In particular, when A and B are both finite-dimensional, α_* can be identified with the partial multiplicities matrix of α.
2. K₀ respects direct limits. If A = ∪_nα_n(A_n)⁻, then K₀(A) is the direct limit ∪_nα_n*(K₀(A_n)).

The dimension group

Since M_∞(M_∞(A)) is isomorphic to M_∞(A), K₀ can only distinguish AF algebras up to stable isomorphism. For example, M₂ and M₄ are not isomorphic but stably isomorphic; K₀(M₂) = K₀(M₄) = Z.

A finer invariant is needed to detect isomorphism classes. For an AF algebra A, we define the scale of K₀(A), denoted by Γ(A), to be the subset whose elements are represented by projections in A:

${\displaystyle \Gamma (A)=\{[p]\,|\,p^{*}=p^{2}=p\in A\}.}$

When A is unital with unit 1_A, the K₀ element [1_A] is the maximal element of Γ(A) and in fact,

${\displaystyle \Gamma (A)=\{x\in K_{0}(A)\,|\,0\leq x\leq [1_{A}]\}.}$

The triple (K₀, K₀⁺, Γ(A)) is called the dimension group of A. If A = M_s, its dimension group is (Z, Z⁺, {1, 2,..., s}).

A group homomorphism between dimension group is said to be contractive if it is scale-preserving. Two dimension group are said to be isomorphic if there exists a contractive group isomorphism between them.

The dimension group retains the essential properties of K₀:

1. A *-homomorphism α : A → B between AF algebras in fact induces a contractive group homomorphism α_* on the dimension groups. When A and B are both finite-dimensional, corresponding to each partial multiplicities matrix ψ, there is a unique, up to unitary equivalence, *-homomorphism α : A → B such that α_* = ψ.
2. If A = ∪_nα_n(A_n)⁻, then the dimension group of A is the direct limit of those of A_n.

Elliott's theorem

Commutative diagrams for Elliott's theorem.

Elliott's theorem says that the dimension group is a complete invariant of AF algebras: two AF algebras A and B are isomorphic if and only if their dimension groups are isomorphic.

Two preliminary facts are needed before one can sketch a proof of Elliott's theorem. The first one summarizes the above discussion on finite-dimensional C*-algebras.

Lemma For two finite-dimensional C*-algebras A and B, and a contractive homomorphism ψ: K₀(A) → K₀(B), there exists a *-homomorphism φ: A → B such that φ_* = ψ, and φ is unique up to unitary equivalence.

The lemma can be extended to the case where B is AF. A map ψ on the level of K₀ can be "moved back", on the level of algebras, to some finite stage in the inductive system.

Lemma Let A be finite-dimensional and B AF, B = (∪_nB_n)⁻. Let β_m be the canonical homomorphism of B_m into B. Then for any a contractive homomorphism ψ: K₀(A) → K₀(B), there exists a *-homomorphism φ: A → B_m such that β_m* φ_* = ψ, and φ is unique up to unitary equivalence in B.

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 β_n* K₀ (B_n).

Theorem (Elliott) Two AF algebras A and B are isomorphic if and only if their dimension groups (K₀(A), K₀⁺(A), Γ(A)) and (K₀(B), K₀⁺(B), Γ(B)) are isomorphic.

The crux of the proof has become known as Elliott's intertwining argument. 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.

Let Φ: (K₀(A), K₀⁺(A), Γ(A)) → (K₀(B), K₀⁺(B), Γ(B)) be a dimension group isomorphism.

1. Consider the composition of maps Φ α_1* : K₀(A₁) → K₀(B). By the previous lemma, there exists B₁ and a *-homomorphism φ₁: A₁ → B₁ such that the first diagram on the right commutes.
2. Same argument applied to β_1* Φ⁻¹ shows that the second diagram commutes for some A₂.
3. Comparing diagrams 1 and 2 gives diagram 3.
4. Using the property of the direct limit and moving A₂ further down if necessary, we obtain diagram 4, a commutative triangle on the level of K₀.
5. For finite-dimensional algebras, two *-homomorphisms induces the same map on K₀ if and only if they are unitary equivalent. So, by composing ψ₁ with a unitary conjugation if needed, we have a commutative triangle on the level of algebras.
6. By induction, we have a diagram of commuting triangles as indicated in the last diagram. The map φ: A → B is the direct limit of the sequence {φ_n}. Let ψ: B → A is the direct limit of the sequence {ψ_n}. It is clear that φ and ψ are mutual inverses. Therefore, A and B are isomorphic.

Furthermore, on the level of K₀, the adjacent diagram commutates for each k. By uniqueness of direct limit of maps, φ_* = Φ.

The Effros-Handelman-Shen theorem

The dimension group of an AF algebra is a Riesz group. The Effros-Handelman-Shen theorem says the converse is true. Every Riesz group, with a given scale, arises as the dimension group of some AF algebra. This specifies the range of the classifying functor K₀ for AF-algebras and completes the classification.

Riesz groups

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, y_i ≥ 0 and x ≤ Σ y_i, then there exists x_i ≥ 0 such that x = Σ x_i, and x_i ≤ y_i 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 Z^k 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 Z^k 's, each with the canonical order structure. This hinges on the following technical lemma, sometimes referred to as the Shen criterion in the literature.

The Shen criterion.

Lemma Let (G, G⁺) be a Riesz group, ϕ: (Z^k, Z^k₊) → (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

${\displaystyle (G,G^{+})=\varinjlim (\mathbb {Z} ^{n_{k}},\mathbb {Z} _{+}^{n_{k}}),}$

where all the connecting homomorphisms in the directed system on the right hand side are positive.