39.02.03 · operator-algebras / af-algebras-k-theory

The Six-Term Exact Sequence, Bott Periodicity, and AF Classification

shipped3 tiersLean: none

Anchor (Master): Blackadar *K-Theory for Operator Algebras* (MSRI 5) Ch. 8-10; Davidson Ch. VII-IX; Elliott (1976); Kirchberg-Phillips (2000)

Intuition Beginner

Cut an operator algebra into two pieces: a small inside part, called an ideal, and the outside view you get after you forget the inside, called the quotient. Each piece has the two fingerprints from before — a count of projections and a count of loops of reversible elements. The question is how the three sets of fingerprints, for the inside, the whole, and the outside, fit together.

The answer is that they chain into a loop of six stations. You can walk a projection-count from the inside to the whole to the outside, then around a corner into the loop-counts, around those three the same way, and back to the start. At each station, what dies on the way in is exactly what was born just before: nothing is lost or doubled. This single six-station loop is the calculator of the whole subject. If you know five of the six counts and the maps between them, you can read off the sixth.

The deepest surprise is that the loop only needs six stations and never more. Counting projections and counting loops are the only two fingerprints there ever are: a third would just repeat the first. That two-step rhythm is called periodicity, and it is why the chain closes up instead of running on forever.

Visual Beginner

The three pieces of a cut algebra share their fingerprints around a closed loop of six stations.

The dictionary reads: the top three boxes count projections for the inside, the whole, and the outside; the bottom three count loops of reversible elements for the same three pieces; the two corner arrows, called the index and the exponential, are the only new maps; and the word exact on every arrow means the chain wastes nothing. Walk all the way around and you return to where you began.

Worked example Beginner

Take the bookshelf algebra of the one-way shift from before, called the Toeplitz algebra. Its inside ideal is the small operators that touch only finitely much of the shelf, and its outside quotient is the functions on a circle. We will read off one corner of the six-station loop.

The outside, functions on a circle, has loops counted by winding: the function that wraps the circle once around counts as $1$ , twice as $2$ , and so on, giving the whole numbers. The inside, small operators, has projections counted by rank, also the whole numbers. The corner arrow called the index carries a wound-up outside loop to an inside projection-count.

What does it carry the once-around wrap to? The once-around wrap lifts to the shift itself, and the shift fails to be reversible by exactly one missing slot — a single rank-one projection. So the index arrow sends $1$ to $1$ (up to a sign): wrapping once costs one missing slot.

What this tells us: the corner of the loop for the bookshelf algebra is the rule "winding number in, missing-slot count out," and it sends the generator $1$ to the generator $1$ . That one identity is the Fredholm index, and the whole six-station loop is just this kind of bookkeeping done for every cut at once.

Check your understanding Beginner

Exercise (easy, multiple choice).

When you cut an algebra into an inside ideal and an outside quotient, how do the three sets of fingerprints fit together?

A. They are completely independent of one another B. They chain into a closed loop of six stations where nothing is lost or doubled C. Only the inside fingerprints matter D. They form an endless chain that never closes

Hint

The picture is a hexagon: three projection-counts on top, three loop-counts on the bottom, joined by two corner arrows.

Answer

B. They chain into a closed loop of six stations.

Feedback-correct: correct; this is the six-term exact sequence, and exactness at each station means what arrives is exactly what the previous arrow could not carry. Feedback-wrong: the whole point is that the three pieces are tightly linked, and the chain closes after six stations because there are only two fingerprints.

Exercise (easy, multiple choice).

Why does the chain of fingerprints close after six stations rather than running on forever?

A. Because the algebra is finite-dimensional B. Because there are only two kinds of fingerprint, and they repeat with a two-step rhythm C. Because projections cannot be counted past six D. Because the ideal is always empty

Hint

A third fingerprint would just repeat the first; this repeating is called periodicity.

Answer

B. There are only two fingerprints, and they repeat.

Feedback-correct: correct; the projection-count and the loop-count are the only two invariants, and periodicity makes a hypothetical third count equal the first, so the chain folds shut into six stations. Feedback-wrong: the closure has nothing to do with the algebra being finite or with a limit on counting; it is the two-step rhythm of periodicity.

Formal definition Intermediate+

Let $0 \to J ι A π A / J \to 0$ be a short exact sequence of C*-algebras 39.01.04. Operator K-theory assigns to it two connecting maps and a cyclic exact sequence.

The index map $\partial_{1} : K_{1} (A / J) \to K_{0} (J)$ is defined on a unitary $u \in U_{n} (A / J)$ by lifting $diag (u, u^{*}) \in U_{2 n} (A / J)$ to a unitary $w \in U_{2 n} (A)$ , setting $\partial_{1} [u] = [w diag (1_{n}, 0_{n}) w^{*}] - [diag (1_{n}, 0_{n})] \in K_{0} (J)$ 39.02.02. The exponential map $\partial_{0} : K_{0} (A / J) \to K_{1} (J)$ is defined on a projection $p \in M_{n} (A / J)$ by lifting it to a self-adjoint $a \in M_{n} (A)$ with $π (a) = p$ and setting $\partial_{0} [p] = [exp (2 π ia)] \in K_{1} (J)$ ; since $π (exp (2 π ia)) = exp (2 π i p) = 1$ , the unitary $exp (2 π ia)$ lies in $U_{n} (J)$ .

The suspension of $A$ is $S A = C_{0} (R) \otimes A = C_{0} (R, A)$ , the $A$ -valued continuous functions on $R$ vanishing at $\pm \infty$ . The fundamental periodicity result is Bott periodicity: the Bott map $β_{A} : K_{0} (A) \to K_{1} (S A)$ , $[p] \mapsto [t \mapsto exp (2 π i t) p + (1 - p)]$ (using $R \cup {\infty} = T$ ), is a natural isomorphism. Equivalently $K_{i} (S A) ≅ K_{i + 1} (A)$ with indices mod $2$ , so $K_{*}$ is 2-periodic in the suspension.

The six-term exact sequence assembles these. From $0 \to J \to A \to A / J \to 0$ one obtains the cyclic diagram

K_{0} (J) \partial_{1} ⏐ ↑ K_{1} (A / J) ι_{*} π_{*} K_{0} (A) K_{1} (A) π_{*} ι_{*} K_{0} (A / J) ↓ ⏐ \partial_{0} K_{1} (J)

exact at every node ^{[Rørdam-Larsen-Laustsen Ch. 9-12]}. A C*-algebra $A$ is approximately finite-dimensional (AF) if $A = \overline{⋃_{n} F_{n}}$ for an increasing sequence of finite-dimensional C*-subalgebras 39.02.02. Its dimension group is the pointed ordered abelian group $(K_{0} (A), K_{0} (A)_{+}, [1_{A}])$ when $A$ is unital, or $(K_{0} (A), K_{0} (A)_{+}, Σ (A))$ with scale $Σ (A) = {[p] : p a projection in A}$ in general.

Counterexamples to common slips

The index and exponential maps are not induced by $*$ -homomorphisms; they are connecting maps of a homological sequence. A unitary over the quotient need not lift to a unitary over $A$ , and the obstruction to lifting is exactly $\partial_{1} [u] \in K_{0} (J)$ . Treating $\partial_{1}$ as functorial in the naive sense (a map between K-groups coming from an algebra map) loses the whole index-theoretic content.
Bott periodicity is a statement about suspension, not about $A$ itself: $K_{0} (A)$ and $K_{1} (A)$ are generally different. The periodicity $K_{0} (A) ≅ K_{1} (S A) ≅ K_{0} (S^{2} A)$ identifies a group with the K-theory of a doubly-suspended algebra, not $K_{0} (A)$ with $K_{1} (A)$ .
The Elliott invariant for AF algebras is the ordered group $(K_0, K_0_+, \text{scale})$ , not the bare group $K_{0}$ . The CAR algebra $M_{2^{\infty}}$ and the universal UHF algebra $Q$ can have non-isomorphic dimension groups ( $Z [1/2]$ versus $Q$ ) even though as bare groups both are divisible-into rank-one rationals; the order and the embedding of the unit distinguish them.

Key theorem with proof Intermediate+

Theorem (the six-term exact sequence). Let $0 \to J ι A π A / J \to 0$ be a short exact sequence of C-algebras. Then the cyclic diagram above is exact at all six nodes, with the index map $\partial_{1}$ and the exponential map $\partial_{0}$ as the two corner connecting homomorphisms.* ^{[Blackadar Ch. 9]}

Proof. The structural input is that $K_{0}$ is half-exact: for the extension above, the sequence $K_{0} (J) ι_{*} K_{0} (A) π_{*} K_{0} (A / J)$ is exact at the middle term $K_{0} (A)$ . This is proved directly: a class in $ker π_{*}$ is $[p] - [q]$ with $π_{*} [p] = π_{*} [q]$ , so after stabilising $p$ and $q$ are connected over $A / J$ ; lifting the connecting path and correcting by a unitary shows $[p] - [q]$ lies in the image of $K_{0} (J)$ . The reverse inclusion $im ι_{*} \subseteq ker π_{*}$ is functoriality applied to $π \circ ι = 0$ 39.02.02.

Half-exactness of $K_{0}$ together with homotopy invariance and stability propagates to a long exact sequence through the mapping cone. Let $C_{π} = {(a, f) \in A \oplus C_{0} ((0, 1], A / J) : f (1) = π (a)}$ be the mapping cone of $π$ . There is a natural extension $0 \to S A / J \to C_{π} \to A \to 0$ relating $C_{π}$ to the suspension, and the inclusion $J ↪ C_{π}$ induced by $a \mapsto (a, 0)$ is a K-equivalence by an Eilenberg-swindle-style excision argument. Iterating half-exactness along the cone construction yields the infinite long exact sequence

\dots \to K_{0} (S A / J) \to K_{0} (J) \to K_{0} (A) \to K_{0} (A / J) \to \dots

with the maps $K_{0} (S A / J) \to K_{0} (J)$ being the connecting homomorphisms.

Now apply Bott periodicity, the natural isomorphism $β_{A} : K_{0} (A) ≅ K_{1} (S A)$ , equivalently $K_{0} (S B) ≅ K_{1} (B)$ for every $B$ . This collapses the infinite sequence: the term $K_{0} (S A / J)$ becomes $K_{1} (A / J)$ , the term $K_{0} (S J)$ becomes $K_{1} (J)$ , and so on, so the long sequence folds into the cyclic six-term diagram. Under this identification the connecting map $K_{0} (S A / J) \to K_{0} (J)$ becomes the index map $\partial_{1} : K_{1} (A / J) \to K_{0} (J)$ , and the connecting map in the dual position becomes the exponential map $\partial_{0} : K_{0} (A / J) \to K_{1} (J)$ . The explicit lift formulas for $\partial_{0}$ and $\partial_{1}$ above are computations of these connecting maps in the projection and unitary pictures; that $\partial_{1}$ agrees with the index map of 39.02.02 is the verification that the Bott identification carries the abstract connecting map to the unitary-lift formula. Exactness at every node is inherited from the long exact sequence. $□$

Bridge. The six-term sequence builds toward the entire computational machinery of classification and appears again in 39.02.01, where it is the Pimsner-Voiculescu sequence computing $K_{*} (A_{θ}) = Z^{2}$ of the irrational rotation algebra as a crossed product. The foundational reason the chain closes after six terms is exactly Bott periodicity: $K_{*}$ has only two values because suspension is 2-periodic, so this is exactly the algebraic shadow of $π_{n} (U) = π_{n + 2} (U)$ in topology. The index map $\partial_{1}$ generalises the Fredholm index of the Toeplitz extension 39.01.04, and the exponential map $\partial_{0}$ is dual to it; putting these together, the central insight is that a short exact sequence of algebras induces a long exact sequence of K-groups that periodicity folds into a hexagon, and the bridge is that classifying C*-algebras up to the data computable by this hexagon is the Elliott program.

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Compute $K_{0} (O_{n}) = Z / (n - 1)$ for the Cuntz algebra using the relation $1 \sim n \cdot 1$ in $K_{0}$ .

Hint

The Cuntz relations give $1 = \sum_{i} s_{i} s_{i}^{*}$ with each $s_{i} s_{i}^{*} \sim 1$ , so $[1] = n [1]$ in $K_{0}$ .

Answer

Each range projection $p_{i} = s_{i} s_{i}^{*}$ satisfies $p_{i} \sim 1$ via the partial isometry $s_{i}$ ( $s_{i}^{*} s_{i} = 1$ , $s_{i} s_{i}^{*} = p_{i}$ ), and $\sum_{i} p_{i} = 1$ , so in $K_{0} (O_{n})$ one has $[1] = \sum_{i} [p_{i}] = n [1]$ , hence $(n - 1) [1] = 0$ 39.01.04. Since $O_{n}$ is generated K-theoretically by $[1]$ and a separating argument (gauge-action computation, or the six-term sequence for the Toeplitz-Cuntz extension $0 \to K \otimes O_{n} \to E_{n} \to O_{n} \to 0$ ) shows there is no further relation, $K_{0} (O_{n}) = Z / (n - 1) Z$ , generated by $[1]$ . Rubric: full credit for $[1] = n [1]$ and the conclusion $Z / (n - 1)$ .

Exercise 6 (hard, short-answer).

Prove that two unital AF algebras $A, B$ with isomorphic pointed ordered $K_{0}$ -groups, $(K_{0} (A), K_{0} (A)_{+}, [1_{A}]) ≅ (K_{0} (B), K_{0} (B)_{+}, [1_{B}])$ , are isomorphic. (Elliott's theorem; you may use the intertwining argument.)

Hint

Approximate a partial map $F_{n} \to B$ realising the K-theory map on the finite stage, then build a commuting ladder by Elliott's approximate-intertwining.

Answer

Write $A = lim F_{n}$ , $B = lim G_{m}$ with $F_{n}, G_{m}$ finite-dimensional. For finite-dimensional algebras a unital $*$ -homomorphism $F_{n} \to G_{m}$ exists and is determined up to unitary equivalence by the induced positive unital map $K_{0} (F_{n}) \to K_{0} (G_{m})$ on the dimension groups $Z^{k_{n}} \to Z^{l_{m}}$ (the multiplicity matrix) 39.02.02. Given the ordered-group isomorphism $ϕ : K_{0} (A) \to K_{0} (B)$ preserving order and unit, restrict to the image of $K_{0} (F_{n})$ ; by continuity of $K_{0}$ this image lands in $K_{0} (G_{m})$ for some $m$ , defining a positive unital map realised by a $*$ -homomorphism $ψ_{n} : F_{n} \to G_{m}$ . Symmetrically $ϕ^{- 1}$ yields maps $G_{m} \to F_{n^{'}}$ . Elliott's approximate-intertwining lemma assembles these into a commuting ladder of $*$ -homomorphisms $F_{n_{1}} \to G_{m_{1}} \to F_{n_{2}} \to \dots$ that converges (in point-norm) to mutually inverse $*$ -isomorphisms $A ≅ B$ , because the diagrams commute up to unitaries that can be absorbed and made to converge. The induced map on $K_{0}$ is $ϕ$ . Hence $A ≅ B$ . Rubric: full credit for the finite-dimensional realisation of positive maps, continuity to land in a stage, and the approximate intertwining.

Exercise 7 (hard, short-answer).

State the Kirchberg-Phillips classification theorem for the purely infinite simple nuclear case, and explain why the Cuntz algebras $O_{n}$ are classified by their K-theory under it.

Hint

A Kirchberg algebra is separable, nuclear, simple, purely infinite; in the UCT class the invariant is the pair $(K_{0}, K_{1})$ with the class of the unit.

Answer

A Kirchberg algebra is a separable, nuclear, simple, purely infinite C*-algebra. The Kirchberg-Phillips theorem states: two Kirchberg algebras satisfying the universal coefficient theorem (UCT) are isomorphic iff their K-theory data agree — for the unital case, iff there is an isomorphism of the pairs $(K_{0}, [1], K_{1})$ , and stably iff $(K_{0}, K_{1})$ agree ^{[Kirchberg-Phillips 2000]}. The Cuntz algebras $O_{n}$ ( $2 \leq n < \infty$ ) are unital Kirchberg algebras in the UCT class with $K_{0} (O_{n}) = Z / (n - 1)$ , $[1] = 1$ , $K_{1} (O_{n}) = 0$ 39.01.04; since these invariants distinguish all the $O_{n}$ (different $n$ give different $Z / (n - 1)$ ), the classification recovers each $O_{n}$ from its K-theory, and $O_{\infty}$ (with $K_{0} = Z$ , $[1] = 1$ ) is likewise pinned down. The theorem is the purely-infinite counterpart of Elliott's AF classification: ordered $K_{0}$ for stably finite, plain $K_{0} \oplus K_{1}$ for purely infinite. Rubric: full credit for the Kirchberg-algebra definition, the UCT/K-theory invariant, and the $O_{n}$ computation.

Lean formalization Intermediate+

lean_status: none — Mathlib has C*-algebras, ideals, quotients, matrices over them, and the abstract Grothendieck group, but not the suspension functor, the mapping cone, half-exactness, the index and exponential connecting maps, Bott periodicity, the six-term exact sequence, the ordered $K_{0}$ as a complete invariant, or the Elliott and Kirchberg-Phillips classification theorems.

The intended statements read schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.CStarAlgebra.Quotient

variable {J A : Type*} [CStarAlgebra J] [CStarAlgebra A]

/-- Bott periodicity: K₀ of A is K₁ of the suspension SA. -/
theorem bott_periodicity (A : Type*) [CStarAlgebra A] :
    K0 A ≃+ K1 (Suspension A) :=
  sorry  -- Bott map [p] ↦ loop t ↦ exp(2πit)p + (1-p)

/-- The six-term exact sequence of an extension 0 → J → A → A/J → 0. -/
theorem six_term_exact
    (ι : J →⋆ₐ[ℂ] A) (π : A →⋆ₐ[ℂ] (A ⧸ ι.range)) (hext : IsExtension ι π) :
    ExactCyclic6 (K0 J) (K0 A) (K0 (A ⧸ ι.range))
                 (K1 (A ⧸ ι.range)) (K1 A) (K1 J)
                 (indexMap π) (expMap π) :=
  sorry

/-- Elliott: unital AF algebras are classified by pointed ordered K₀. -/
theorem elliott_AF_classification
    (A B : Type*) [CStarAlgebra A] [CStarAlgebra B] [IsAF A] [IsAF B]
    (h : OrderedK0Iso A B) : A ≃⋆ₐ[ℂ] B :=
  sorry

Advanced results Master

Naturality and the mapping cylinder. The six-term sequence is natural in the extension: a morphism of short exact sequences induces a ladder of six-term diagrams commuting with the connecting maps. The cleanest construction realises $\partial_{1}$ and $\partial_{0}$ as honest induced maps via the mapping cone $C_{π}$ and the natural K-equivalence $K_{*} (C_{π}) ≅ K_{*} (S A / J)$ paired with $K_{*} (J) ≅ K_{*} (C_{π})$ by excision; Bott periodicity then identifies the resulting long exact sequence with the hexagon. The exponential map admits the dual description $\partial_{0} = β^{- 1} \circ \partial_{1}^{S} \circ β$ after suspending, which is why it is computed by the exponential $exp (2 π ia)$ — suspending a projection and applying the index produces a unitary loop, and the loop is the exponential of the self-adjoint lift.

The Pimsner-Voiculescu sequence. For a crossed product $A ⋊_{α} Z$ the Pimsner-Voiculescu exact sequence reads $$ K_0(A) \xrightarrow{,1 - \alpha_*,} K_0(A) \to K_0(A \rtimes \mathbb{Z}) \to K_1(A) \xrightarrow{,1 - \alpha_*,} K_1(A) \to K_1(A \rtimes \mathbb{Z}), $$ itself a six-term sequence for the Toeplitz extension of the crossed product (the mapping-torus presentation) 39.01.04. Applied to the irrational rotation algebra $A_{θ} = C (T) ⋊_{θ} Z$ it gives $K_{0} (A_{θ}) = Z^{2}$ , $K_{1} (A_{θ}) = Z^{2}$ , with the order on $K_{0}$ scaled by $1 + Z θ$ — the trace pairing $τ_{*} : K_{0} (A_{θ}) \to R$ has image $Z + Z θ$ , distinguishing the $A_{θ}$ for different $θ$ up to $θ \mapsto 1 - θ$ 39.02.01.

Elliott's classification of AF algebras. A unital AF algebra is determined up to isomorphism by its dimension group $(K_{0} (A), K_{0} (A)_{+}, [1_{A}])$ , and every countable pointed ordered abelian group that is a Riesz group (an ordered group with the Riesz interpolation property and no nonzero infinitesimals, i.e. an unperforated group with interpolation) arises. The proof is the approximate intertwining: positive unital maps between dimension groups lift to $*$ -homomorphisms between finite stages, and Elliott's intertwining lemma assembles a commuting ladder converging to an isomorphism. Thus the classification of AF algebras is the classification of dimension groups — a purely order-theoretic problem — and Effros-Handelman-Shen characterise exactly which ordered groups occur. The UHF algebras correspond to subgroups of $Q$ containing $Z$ with the inherited order, classified by supernatural numbers; the CAR algebra $M_{2^{\infty}}$ is $(Z [1/2], Z [1/2]_{+}, 1)$ .

The Elliott program and Kirchberg-Phillips. Elliott conjectured that all separable nuclear simple C*-algebras are classified by the Elliott invariant $Ell (A) = (K_{0} (A), K_{0} (A)_{+}, [1_{A}], K_{1} (A), T (A), ρ_{A})$ — ordered K-theory, the trace simplex $T (A)$ , and the pairing $ρ_{A} : T (A) \times K_{0} (A) \to R$ . The purely infinite simple nuclear case was settled by Kirchberg and Phillips: Kirchberg algebras in the UCT class are classified by $(K_{0}, K_{1})$ with the unit class ^{[Kirchberg-Phillips 2000]}, with $O_{2}$ ( $K_{*} = 0$ ) absorbing — $A \otimes O_{2} ≅ O_{2}$ for every Kirchberg algebra $A$ — and $O_{\infty}$ tensorially stabilising the purely infinite condition. The stably finite case, after counterexamples of Rørdam and Toms forced the addition of regularity hypotheses, was completed for $Z$ -stable algebras (the Jiang-Su algebra $Z$ being the stably finite analogue of $O_{\infty}$ ), making the Toms-Winter regularity the precise boundary of classifiability.

Synthesis. The six-term sequence is the central insight that organises the entire classification program: it is the foundational reason a short exact sequence of C*-algebras becomes a computable cyclic chain of abelian groups, and Bott periodicity is exactly the reason the chain closes after six terms rather than running on forever, since $K_{*}$ is 2-periodic in the suspension. The index map generalises the Fredholm index of the Toeplitz extension 39.01.04, the exponential map is dual to it, and putting these together with the Pimsner-Voiculescu specialisation gives every concrete K-group of crossed products and extensions — this is exactly the engine that computes the dimension group of an AF algebra and the invariant of the irrational rotation algebra 39.02.01. The bridge from computation to classification is Elliott's theorem: the dimension group is a complete invariant for AF algebras, and the central insight that classifying an entire class of C*-algebras has become computing and comparing ordered K-theory launches the Elliott program, whose purely infinite half is the Kirchberg-Phillips theorem classifying the Cuntz algebras and all Kirchberg algebras by their bare K-theory.

Full proof set Master

Proposition (the exponential map is well-defined). Let $0 \to J \to A π A / J \to 0$ be a short exact sequence and $p \in M_{n} (A / J)$ a projection. Lift $p$ to a self-adjoint $a \in M_{n} (A)$ with $π (a) = p$ (possible since the self-adjoint part surjects). Then $u = exp (2 π ia) \in U_{n} (J)$ because $π (u) = exp (2 π i p) = 1$ (as $p^{2} = p$ gives $exp (2 π i p) = 1 + (e^{2 π i} - 1) p = 1$ ), and $\partial_{0} [p] := [u] \in K_{1} (J)$ is independent of the lift and of the representative of $[p]$ . Indeed two self-adjoint lifts $a, a^{'}$ of $p$ differ by an element of $M_{n} (J)_{s a}$ , so $exp (2 π ia)$ and $exp (2 π i a^{'})$ are connected through $exp (2 π i (a + t (a^{'} - a)))$ , a path in $U_{n} (J)$ , giving the same $K_{1} (J)$ class; additivity follows from $exp (2 π i diag (a, a^{'})) = diag (exp (2 π ia), exp (2 π i a^{'}))$ , so $\partial_{0}$ is a homomorphism. $□$

Proposition (exactness at $K_{0} (A / J)$ ). The composite $K_{0} (A) π_{*} K_{0} (A / J) \partial_{0} K_{1} (J)$ is zero, and $ker \partial_{0} = im π_{*}$ . If $[p] = π_{*} [q]$ for a projection $q \in M_{n} (A)$ , then $q$ is itself a self-adjoint lift of $p$ with $exp (2 π i q) = 1$ (since $q$ is a projection), so $\partial_{0} π_{*} [q] = [1] = 0$ . Conversely, if $\partial_{0} [p] = 0$ then $exp (2 π ia) \in U_{n} (J)_{0}$ for a self-adjoint lift $a$ ; writing $exp (2 π ia) = exp (2 π ib)$ for $b \in M_{n} (J)_{s a}$ via a path, the element $a - b$ is a self-adjoint lift of $p$ with $exp (2 π i (a - b)) = 1$ , whose spectrum lies in $Z$ , hence $a - b$ is a projection-valued lift, exhibiting $[p] \in im π_{*}$ . $□$

Proposition (Bott periodicity by the Toeplitz extension). For any C*-algebra $B$ , $K_{1} (S B) ≅ K_{0} (B)$ . Tensor the Toeplitz extension $0 \to K \to T \to C (T) \to 0$ with $B$ and restrict to functions vanishing at a basepoint, giving $0 \to K \otimes B \to T_{0} \otimes B \to S B \to 0$ with $T_{0}$ the kernel of evaluation. The Toeplitz algebra $T$ is K-equivalent to $C$ (it is homotopy equivalent to $C$ through the path contracting the shift, so $K_{*} (T_{0}) = 0$ ), so the six-term sequence of this extension degenerates and its connecting map $\partial : K_{1} (S B) \to K_{0} (K \otimes B) = K_{0} (B)$ is an isomorphism. This is the Cuntz proof of Bott periodicity: periodicity is the statement that the Toeplitz extension is, after suspension, K-theoretically split into a degree shift. $□$

Proposition (dimension groups are unperforated with interpolation). Let $A = lim F_{n}$ be AF with $F_{n} = ⨁_{i} M_{n_{i}}$ . The ordered group $K_{0} (A) = lim Z^{k_{n}}$ (positive cone the limit of $Z_{\geq 0}^{k_{n}}$ , connecting maps the positive multiplicity matrices) is unperforated — $m x \geq 0$ for some $m \geq 1$ forces $x \geq 0$ — because each $Z^{k_{n}}$ is and positivity is detected at a finite stage. It satisfies Riesz interpolation: given $a_{1}, a_{2} \leq b_{1}, b_{2}$ there is $c$ with $a_{i} \leq c \leq b_{j}$ , inherited from the lattice order on each $Z^{k_{n}}$ and preserved in the limit. By the Effros-Handelman-Shen theorem these two properties characterise dimension groups exactly: every countable unperforated ordered group with interpolation is the dimension group of some AF algebra. $□$

Proposition ( $K_{0} (O_{n}) = Z / (n - 1)$ via the six-term sequence). The Toeplitz-Cuntz extension $0 \to K \otimes O_{n} \to E_{n} \to O_{n} \to 0$ (the algebra $E_{n}$ generated by $n$ isometries with orthogonal ranges, no partition relation) has $E_{n}$ K-equivalent to $C$ and the inclusion-induced map on $K_{0}$ equal to multiplication by $1 - n$ (the $n$ range projections each contribute $[1]$ , and the relation imposes $\sum [p_{i}] = n [1]$ against $[1]$ ). The six-term sequence then reads $K_{0} (K \otimes O_{n}) = Z 1 - n Z = K_{0} (E_{n}) \to K_{0} (O_{n}) \to 0$ with $K_{1}$ terms vanishing, so $K_{0} (O_{n}) = coker (1 - n : Z \to Z) = Z / (n - 1) Z$ and $K_{1} (O_{n}) = ker (1 - n) = 0$ for $n \geq 2$ . $□$

Connections Master

Operator K-Theory: K_0 and K_1 of C-Algebras 39.02.02* — the index map $\partial_{1}$ and exponential map $\partial_{0}$ assembled here are the connecting homomorphisms whose unitary-lift and projection-lift formulas were established there; the four pillars (functoriality, stability, homotopy invariance, half-exactness) are exactly the input that, with Bott periodicity, forces the cyclic six-term sequence.
The Toeplitz Algebra, Cuntz Algebras, and Extensions 39.01.04 — the Toeplitz extension is the worked example computing $\partial_{1} =$ Fredholm index $= -$ winding, and the Cuntz relations give $K_{0} (O_{n}) = Z / (n - 1)$ via the six-term sequence; these extensions are the concrete inputs on which the abstract hexagon is exercised.
AF Algebras, Bratteli Diagrams, and the Irrational Rotation Algebra 39.02.01 — Elliott's classification completes here as a corollary: the dimension group $(K_0, K_0_+, \text{scale})$ is a complete invariant for AF algebras, and the Pimsner-Voiculescu specialisation of the six-term sequence computes $K_{*} (A_{θ}) = Z^{2}$ of the irrational rotation algebra as a crossed product.
Comparison of Projections and the Murray-von Neumann Type Classification 39.03.04 — the stably finite versus purely infinite dichotomy that splits the classification into the Elliott (ordered $K_{0}$ ) and Kirchberg-Phillips (bare $K_{0} \oplus K_{1}$ ) regimes is the K-theoretic image of the finite/infinite dichotomy of the type classification.
Topological K-theory 03.08.01 — Bott periodicity for C*-algebras is the noncommutative extension of the topological Bott periodicity $π_{n} (U) = π_{n + 2} (U)$ ; the suspension $S A = C_{0} (R) \otimes A$ is the operator-algebraic reduced suspension, and the six-term sequence generalises the long exact sequence of a cofibration in topological K-theory.

Historical & philosophical context Master

The six-term exact sequence emerged in the 1970s as operator K-theory matured from the Atiyah-Hirzebruch topological theory into a tool for C*-algebras. Bott periodicity, proved topologically by Bott in 1957 for the unitary group, was transported to the operator-algebraic setting; Cuntz gave a short proof in 1981 using the Toeplitz extension, replacing the analytic machinery with the algebra of a single isometry ^{[Cuntz 1981]}. The index map identified with the Fredholm index through the Toeplitz extension tied the sequence to the Atiyah-Singer index theorem and to Brown-Douglas-Fillmore extension theory.

Elliott proved in 1976 that AF algebras are classified by their pointed ordered $K_{0}$ , the dimension group ^{[Elliott 1976]}, reducing a problem about operator algebras to one about ordered abelian groups; Effros, Handelman, and Shen characterised the groups that arise. This was the seed of the Elliott program, the conjecture that separable nuclear simple C*-algebras are classified by K-theory and traces. The purely infinite case was settled by Kirchberg and Phillips around 2000 ^{[Kirchberg-Phillips 2000]}, with $O_{2}$ absorption and $O_{\infty}$ stabilisation as the structural pillars. Counterexamples of Rørdam and Toms in the 2000s showed the stably finite conjecture needed regularity hypotheses, and the program reached its modern form with $Z$ -stability and the Toms-Winter conjecture, placing the Jiang-Su algebra as the stably finite analogue of $O_{\infty}$ .

Bibliography Master

Blackadar, B., K-Theory for Operator Algebras, 2nd ed., MSRI Publications 5, Cambridge University Press, 1998.
Rørdam, M., Larsen, F., and Laustsen, N. J., An Introduction to K-Theory for C-Algebras*, London Mathematical Society Student Texts 49, Cambridge University Press, 2000.
Davidson, K. R., C-Algebras by Example*, Fields Institute Monographs 6, American Mathematical Society, 1996. Ch. VII-IX.
Elliott, G. A., "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras", Journal of Algebra 38 (1976), 29-44.
Effros, E. G., Handelman, D. E., and Shen, C.-L., "Dimension groups and their affine representations", American Journal of Mathematics 102 (1980), 385-407.
Cuntz, J., "K-theory for certain C*-algebras", Annals of Mathematics 113 (1981), 181-197.
Pimsner, M. and Voiculescu, D., "Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras", Journal of Operator Theory 4 (1980), 93-118.
Kirchberg, E. and Phillips, N. C., "Embedding of exact C*-algebras in the Cuntz algebra O_2", Journal für die reine und angewandte Mathematik 525 (2000), 17-53.
Rørdam, M., "A simple C*-algebra with a finite and an infinite projection", Acta Mathematica 191 (2003), 109-142.

Operator-algebras spine, third unit of the AF-algebras / K-theory chapter. Produced as the computational engine: the index and exponential connecting maps, Bott periodicity $K_{0} (A) ≅ K_{1} (S A)$ and the resulting 2-periodicity, the cyclic six-term exact sequence of an extension, the Toeplitz extension as the worked index computation, the Pimsner-Voiculescu specialisation, Elliott's classification of AF algebras by the dimension group completed as a corollary, and the Kirchberg-Phillips classification of Kirchberg algebras and the Elliott program.

Prerequisites

39.02.02
39.01.04

Tier anchors

beginner: Strogatz-style 'cut an algebra into an ideal and a quotient, and watch the projections and loops flow around a six-station loop'; the cut-and-trace picture of an exact sequence
intermediate: Rørdam-Larsen-Laustsen *An Introduction to K-Theory for C*-Algebras* Ch. 8-12; Davidson *C*-Algebras by Example* Ch. VII-IX
master: Blackadar *K-Theory for Operator Algebras* (MSRI 5) Ch. 8-10; Davidson Ch. VII-IX; Elliott (1976); Kirchberg-Phillips (2000)

References

Rørdam, M., Larsen, F., Laustsen, N. J. — An Introduction to K-Theory for C*-Algebras · London Math. Soc. Student Texts 49 (2000), Ch. 8-12: the index and exponential maps, Bott periodicity, the six-term exact sequence, classification
Blackadar, B. — K-Theory for Operator Algebras · MSRI Publications 5, 2nd ed. (1998), Ch. 8-10: cyclic six-term sequence, Bott periodicity for C*-algebras, the Elliott classification of AF algebras
Davidson, K. R. — C*-Algebras by Example · Fields Institute Monographs 6 (1996), Ch. VII-IX: the six-term sequence, AF classification by dimension groups, extensions
Elliott, G. A. — On the classification of inductive limits of sequences of semisimple finite-dimensional algebras · J. Algebra 38 (1976), 29-44: AF algebras classified by the pointed ordered K_0 (dimension group)
Kirchberg, E. and Phillips, N. C. — Embedding of exact C*-algebras in the Cuntz algebra O_2 · J. reine angew. Math. 525 (2000): classification of Kirchberg algebras by K-theory; the Cuntz algebras O_n

Estimated time

beginner: 18m
intermediate: 55m
master: 95m