# Approximately finite-dimensional C*-algebra

In mathematics, an **approximately finite-dimensional (AF) 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 *K*_{0} 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 *K*_{0} 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.

The counterpart of simple AF C*-algebras in the von Neumann algebra world are the hyperfinite factors, which were classified by Connes and Haagerup.

In the context of noncommutative geometry and topology, AF C*-algebras are noncommutative generalizations of *C*_{0}(*X*), where *X* is a totally disconnected metrizable space.

## Definition and basic properties

### Finite-dimensional C*-algebras

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

where *M _{i}* denotes the full matrix algebra of

*i*×

*i*matrices.

Up to unitary equivalence, a unital *-homomorphism Φ : *M _{i}* →

*M*is necessarily of the form

_{j}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

is specified, up to unitary equivalence, by a *t* × *s* matrix of *partial multiplicities* (*r*_{l k}) satisfying, for all *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*and the number of arrows from

_{l}*n*to

_{k}*m*is the partial multiplicity

_{l}*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:

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

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.

- (A Bratteli diagram of the CAR algebra)

If an AF algebra *A* = (∪_{n}*A _{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 ∪_{n}*A _{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]

## Classification

### K_{0}

_{0}

The K-theoretic group *K _{0}* 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*,

*K*

_{0}(

*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

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

where ⊕ is the orthogonal direct sum. This makes *P*(*A*)/~ a semigroup that has the cancellation property. We denote this semigroup by *K _{0}*(

*A*)

^{+}. Performing the Grothendieck group construction gives an abelian group, which is

*K*(

_{0}*A*).

*K _{0}*(

*A*) carries a natural order structure: we say [

*p*] ≤ [

*q*] if

*p*is Murray-von Neumann equivalent to a subprojection of

*q*. This makes

*K*(

_{0}*A*) an ordered group whose positive cone is

*K*(

_{0}*A*)

^{+}.

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

one has

Two essential features of the mapping *A* ↦ *K*_{0}(*A*) are:

*K*_{0}is a (covariant) functor. A *-homomorphism*α*:*A*→*B*between AF algebras induces a group homomorphism*α*_{*}:*K*_{0}(*A*) →*K*_{0}(*B*). In particular, when*A*and*B*are both finite-dimensional,*α*_{*}can be identified with the partial multiplicities matrix of*α*.*K*_{0}respects direct limits. If*A*= ∪_{n}*α*(_{n}*A*)_{n}^{−}, then*K*_{0}(*A*) is the direct limit ∪_{n}*α*_{n*}(*K*_{0}(*A*_{n})).

### The dimension group

Since *M*_{∞}(*M*_{∞}(*A*)) is isomorphic to *M*_{∞}(*A*), *K*_{0} can only distinguish AF algebras up to *stable isomorphism*. For example, *M*_{2} and *M*_{4} are not isomorphic but stably isomorphic; *K*_{0}(*M*_{2}) = *K*_{0}(*M*_{4}) = **Z**.

A finer invariant is needed to detect isomorphism classes. For an AF algebra *A*, we define the **scale** of *K*_{0}(*A*), denoted by Γ(*A*), to be the subset whose elements are represented by projections in *A*:

When *A* is unital with unit 1_{A}, the *K*_{0} element [1_{A}] is the maximal element of Γ(*A*) and in fact,

The triple (*K*_{0}, *K*_{0}^{+}, 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 *K*_{0}:

- 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*. - If
*A*= ∪_{n}*α*(_{n}*A*)_{n}^{−}, then the dimension group of*A*is the direct limit of those of*A*._{n}

### 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*: *K*_{0}(*A*) → *K*_{0}(*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 *K*_{0} 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* = (∪_{n}*B _{n}*)

^{−}. Let

*β*be the canonical homomorphism of

_{m}*B*into

_{m}*B*. Then for any a contractive homomorphism

*ψ*:

*K*

_{0}(

*A*) →

*K*

_{0}(

*B*), there exists a *-homomorphism

*φ*:

*A*→

*B*such that

_{m}*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 *K*_{0}(*A*) is finitely generated and, since *K*_{0} respects direct limits, *K*_{0}(*B*) = ∪_{n} *β _{n*}*

*K*

_{0}(

*B*).

_{n}**Theorem (Elliott)** Two AF algebras *A* and *B* are isomorphic if and only if their dimension groups (*K*_{0}(*A*), *K*_{0}^{+}(*A*), Γ(*A*)) and (*K*_{0}(*B*), *K*_{0}^{+}(*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: (*K*_{0}(*A*), *K*_{0}^{+}(*A*), Γ(*A*)) → (*K*_{0}(*B*), *K*_{0}^{+}(*B*), Γ(*B*)) be a dimension group isomorphism.

- Consider the composition of maps Φ
*α*_{1*}:*K*_{0}(*A*_{1}) →*K*_{0}(*B*). By the previous lemma, there exists*B*_{1}and a *-homomorphism*φ*_{1}:*A*_{1}→*B*_{1}such that the first diagram on the right commutes. - Same argument applied to
*β*_{1*}Φ^{−1}shows that the second diagram commutes for some*A*_{2}. - Comparing diagrams 1 and 2 gives diagram 3.
- Using the property of the direct limit and moving
*A*_{2}further down if necessary, we obtain diagram 4, a commutative triangle on the level of*K*_{0}. - For finite-dimensional algebras, two *-homomorphisms induces the same map on
*K*_{0}if and only if they are unitary equivalent. So, by composing*ψ*_{1}with a unitary conjugation if needed, we have a commutative triangle on the level of algebras. - 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*}. Let_{n}*ψ*:*B*→*A*is the direct limit of the sequence {*ψ*}. It is clear that_{n}*φ*and*ψ*are mutual inverses. Therefore,*A*and*B*are isomorphic.

Furthermore, on the level of *K*_{0}, 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 *K*_{0} 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*≤ ∑

*y*, then there exists

_{i}*x*≥ 0 such that

_{i}*x*= ∑

*x*, and

_{i}*x*≤

_{i}*y*for each

_{i}*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, (*K*_{0}, *K*_{0}^{+}) 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 (*K*_{0}, *K*_{0}^{+}) 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.

**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

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

#### The theorem

**Theorem** If (*G*, *G*^{+}) is a countable Riesz group with scale Γ(*G*), then there exists an AF algebra *A* such that (*K*_{0}, *K*_{0}^{+}, Γ(*A*)) = (*G*, *G*^{+}, Γ(*G*)). In particular, if Γ(*G*) = [0, *u _{G}*] with maximal element

*u*, then

_{G}*A*is unital with [1

_{A}] = [

*u*].

_{G}Consider first the special case where Γ(*G*) = [0, *u _{G}*] with maximal element

*u*. Suppose

_{G}Dropping to a subsequence if necessary, let

where *φ*_{1}(*u*_{1}) = *u _{G}* for some element

*u*

_{1}. Now consider the order ideal

*G*

_{1}generated by

*u*

_{1}. Because each

*H*

_{1}has the canonical order structure,

*G*

_{1}is a direct sum of

**Z**'s (with the number of copies possible less than that in

*H*

_{1}). So this gives a finite-dimensional algebra

*A*

_{1}whose dimension group is (

*G*

_{1}

*G*

_{1}

^{+}, [0,

*u*

_{1}]). Next move

*u*

_{1}forward by defining

*u*

_{2}=

*φ*

_{12}(

*u*

_{1}). Again

*u*

_{2}determines a finite-dimensional algebra

*A*

_{2}. There is a corresponding homomorphism

*α*

_{12}such that

*α*

_{12*}= φ

_{12}. Induction gives a directed system

whose *K*_{0} is

with scale

This proves the special case.

A similar argument applies in general. Observe that the scale is by definition a directed set. If Γ(*G*) = {*v _{k}*}, one can choose

*u*∈ Γ(

_{k}*G*) such that

*u*≥

_{k}*v*

_{1}...

*v*. The same argument as above proves the theorem.

_{k}## Examples

By definition, uniformly hyperfinite algebras are AF and unital. Their dimension groups are the subgroups of **Q**. For example, for the 2 × 2 matrices *M*_{2}, *K*_{0}(*M*_{2}) is the group of rational numbers of the form *a*/2 for *a* in **Z**. The scale is Γ(*M*_{2}) = {0, ½, 1}. For the CAR algebra *A*, *K*_{0}(*A*) is the group of dyadic rationals with scale *K*_{0}(*A*) ∩ [0, 1], with 1 = [1_{A}]. All such groups are simple, in a sense appropriate for ordered groups. Thus UHF algebras are simple C*-algebras. In general, the groups which are not dense in **Q** are the dimension groups of *M _{k}* for some

*k*.

Commutative C*-algebras, which were characterized by Gelfand, are AF precisely when the spectrum is totally disconnected.[2] The continuous functions *C*(*X*) on the Cantor set *X* is one such example.

## Elliott's classification program

It was proposed by Elliott that other classes of C*-algebras may be classifiable by K-theoretic invariants. For a C*-algebra *A*, the *Elliott invariant* is defined to be

where *T*^{+}(*A*) is the tracial positivel linear functionals in the weak-* topology, and *ρ _{A}* is the natural pairing between

*T*

^{+}(

*A*) and

*K*

_{0}(

*A*).

The original conjecture by Elliott stated that the Elliott invariant classifies simple unital separable nuclear C*-algebras.

In the literature, one can find several conjectures of this kind with corresponding modified/refined Elliott invariants.

## Von Neumann algebras

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 II_{1} factor. Connes obtained the analogous result for the II_{∞} factor. Powers exhibited a family of non-isomorphic type III hyperfinite factors with cardinality of the continuum. Today we have a complete classification of hyperfinite factors.

## Notes

- Lawrence G. Brown. Extensions of AF Algebras: The Projection Lifting Problem. Operator Algebras and Applications, Proceedings of symposia in pure mathematics, vol. 38, Part 1, pp. 175-176, American Mathematical Soc., 1982
- Davidson 1996, p. 77.

## References

- Bratteli, O. (1972),
*Inductive limits of finite dimensional C*-algebras*, Trans. Amer. Math. Soc.**171**, 195-234. - Davidson, K.R. (1996),
*C*-algebras by Example*, Field Institute Monographs**6**, American Mathematical Society. - Effros, E.G., Handelman, D.E., and Shen C.L. (1980),
*Dimension groups and their affine representations*, Amer. J. Math.**102**, 385-402. - Elliott, G.A. (1976),
*On the Classification of Inductive Limits of Sequences of Semisimple Finite-Dimensional Algebras*, J. Algebra**38**, 29-44. - Elliott, G.A. and Toms, A.S. (2008),
*Regularity properties in the classification program for separable amenable C-algebras*, Bull. Amer. Math. Soc.**45**, 229-245. - Fillmore, P.A.(1996),
*A User's Guide for Operator Algebras*, Wiley-Interscience. - Rørdam, M. (2002),
*Classification of Nuclear C*-Algebras*, Encyclopaedia of Mathematical Sciences**126**, Springer-Verlag.

## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "AF-algebra",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4