Approximately finite dimensional C*-algebra

In C*-algebras, an approximately finite dimensional, or AF, C*-algebra is one that is the inductive limit of a sequence of finite dimensional C*-algebras. Approximate finite dimensionality was first defined and described combinatorially by Bratteli. Elliott gave a complete classification of AF algebras using the "K"₀ functor whose range consists of certain type of abelian groups.

Definition and basic properties

Finite dimensional C*-algebras

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

:$oplus\; \_k\; M\_\{n\_k\},$

where "M_i" denotes the full matrix algabra of "i" × "i" matrices.

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

:$Phi\; (a)\; =\; a\; otimes\; I\_r,$

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

:$Phi:\; oplus\; \_1\; ^s\; M\_\{n\_k\}\; ightarrow\; oplus\; \_1\; ^t\; M\_\{m\_l\}$

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

:$sum\_k\; r\_\{l\; k\}\; n\_k\; =\; m\_l.;$

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:

:$A\; =\; varinjlim\; cdots\; ightarrow\; A\_i\; ,\; stackrel\{alpha\_i\}\{\; ightarrow\}\; A\_\{i+1\}\; ightarrow\; cdots\; ,$

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

:$A\; =\; overline\; \{cup\_n\; A\_n\}.$

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 give on the right. The two arrows between nodes means each connecting map is an embedding of multiplicity 2.

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".

The extension of an AF algebra by another AF algebra is again AF.

Classification

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 "A". "M_n"("A") can be embedded into "M"_{"n" + 1}("A") canonically, into the "upper left corner". Consider the algebraic direct limit

:$M\; \_\{infty\}\; (A)\; =\; varinjlim\; cdots\; ightarrow\; M\_n(A)\; ightarrow\; M\_\{n+1\}(A)\; ightarrow\; cdots\; .$

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

:$[p]\; +\; [q]\; =\; [p\; oplus\; q]$

where ⊕ is the orthogonal direct sum. This makes "P"("A")/~ a semigroup that has the cancellation property. We denote this semigroup by "K₀"("A")⁺. Performing the Grothendieck 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

:$A\; =\; oplus\; \_\{k\; =\; 1\}\; ^m\; M\_\{n\_k\},$

one has

:$(K\_0(A),\; K\_0(A)^+)\; =\; (mathbb\{Z\}^m,\; mathbb\{Z\}\_+\; ^m).$

Two essential features of the mapping "A" $mapsto$ "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 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":

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

When "A" is unital with unit 1_"A", the "K"₀ element [1_"A"] is the maximal element of Γ("A").

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_m" be the canonical homomorphism of "B_m" into "B". Then for any a contractive homomorphism "ψ": "K"₀("A") → "K"₀("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 "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 "α"_1* Φ : "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_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 "K"₀, the diagram on the left 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_i", then there exists "x_i" ≥ 0 such that "x" = ∑ "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, ("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 diagram to the right, such that ker("σ") = ker("φ").

Corollary Every Riesz group ("G", "G"⁺) can be expressed as a direct limit

:$(G,\; G^+)\; =\; varinjlim\; (mathbb\{Z\}^\{n\_k\},\; \{Z\}^\{n\_k\}\_+)\; ,$

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_G", then "A" is unital with [1_A] = ["u_G"] .

Consider first the special case where Γ("G") = [0, "u_G"] with maximal element "u_G". Suppose

:$(G,\; G^+)\; =\; varinjlim\; (H\_k,\; H\_k^+)\; ,\; quad\; mbox\{where\}\; quad\; (H,\; H\_k^+)\; =\; (mathbb\{Z\}^\{n\_k\},\; mathbb\{Z\}^\{n\_k\}\_+).$

Dropping to a subsequence if necessary, let

:$Gamma(H\_1)\; =\; \{\; v\; in\; H\_1^+\; |\; phi\_1(v)\; in\; Gamma(G)\; \},$

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* = φ₁₂. Induction gives a directed system

:$A\; =\; varinjlim\; A\_k\; ,$

whose "K"₀ is

:$varinjlim\; (G\_k,\; G\_k^+),$

with scale

:$cup\_k\; phi\_k\; [0,\; u\_k]\; =\; [0,\; u\_G]\; .$

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_k" ∈ Γ("G") such that "u_k" ≥ "v"₁ ... "v_k". The same argument as above proves the theorem.

Examples

By definition, uniformly hyperfinite algebras are AF and unital. Their dimension groups are the countable subgroups of R. For example, for the 2 × 2 matrices "M"₂, "K"₀("M"₂) is Z [½] , the rational numbers of the form "a"/2. The scale is Γ("M"₂) = Z [½] ∩ [0, 1] = [0, ½, 1] . For the CAR algebra "A", "K"₀("A") is the 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 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. The continuous functions "C"("X") on the Cantor set "X" is one such example. Its "K"₀ group is Z^c, where c is the cardinality of the continuum. The scale is [0, 1] ^c.

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

:$mbox\{Ell\}(A)\; ;\; stackrel\{mbox\{def\{=\};\; (;\; (K\_0(A),\; K\_0(A)^+,\; Gamma(A)\; ),\; K\_1(A),\; T^+(A),\; ho\_A\; ;),$

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 Elliott type, 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 II₁ 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.

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 transformations", Amer. J. Math. 102, 385-402.

*Elliott, G.A. (1976), "On the classification of inductive limits of sequences of semi-simple 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.