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

Operator K-Theory: K_0 and K_1 of C*-Algebras

shipped3 tiersLean: none

Anchor (Master): Blackadar *K-Theory for Operator Algebras* (MSRI 5) Ch. 1-9; Davidson Ch. V-VII; Wegge-Olsen *K-Theory and C*-Algebras*

Intuition Beginner

Inside an algebra of operators there are special elements called projections: things that square to themselves, the operator version of "select this part, ignore the rest." Two projections should count as the same if one can be slid onto the other by a rotation inside the algebra. So a natural question is: how many genuinely different sizes of projection can you build? Adding projections that live in different corners just stacks their sizes, so the sizes form a system you can add. Allowing formal differences — the size of one projection minus another — turns that system into a group. This group is called $K_{0}$ , and it is the first fingerprint of the algebra.

There is a second fingerprint. The reversible elements of the algebra, the unitaries, can sometimes be slid continuously back to the identity and sometimes cannot: they can be caught in a loop that will not untwist. Counting the genuinely different loops gives a second group, called $K_{1}$ . Together $K_{0}$ and $K_{1}$ are the two simplest numerical shadows an operator algebra casts, and they are strong enough to tell many algebras apart even when nothing else will.

The surprise is that these same two groups, for an algebra of continuous functions on a shape, recover the count of vector bundles on that shape. So operator algebras carry their own topology.

Visual Beginner

Projections build $K_{0}$ by stacking and allowing differences; unitaries build $K_{1}$ by counting loops that will not untwist.

The dictionary reads: a projection is a corner you can select; sliding one corner onto another by a rotation inside the algebra makes them count as equal; stacking corners that sit in different places adds their sizes; and a formal difference of sizes completes the count into the group $K_{0}$ . On the right, a unitary that can be walked back to the identity counts as nothing, while one trapped in a loop counts as a generator of $K_{1}$ .

Worked example Beginner

Take the simplest interesting algebra: the $2 \times 2$ complex matrices, written $M_{2}$ . What projections can you build, and what sizes do they come in?

A projection here is a matrix $p$ with $p^{2} = p$ and $p = p^{*}$ . The smallest nonzero one selects a single line, like

p = (1000),

which has rank $1$ . The identity matrix selects both lines and has rank $2$ . The zero matrix has rank $0$ . Two projections inside $M_{2}$ count as the same exactly when they have the same rank, because you can rotate one line onto any other line. So the available sizes are $0, 1, 2$ — just the rank.

When you stack projections from larger matrix blocks the rank keeps climbing through all the whole numbers, and allowing formal differences fills in the negatives. The reachable sizes become every integer.

What this tells us: for $M_{2}$ the group $K_{0}$ is the whole numbers under addition, $Z$ , with the size of a projection read off as its rank. Counting projections has become counting integers.

Check your understanding Beginner

Formal definition Intermediate+

Let $A$ be a C*-algebra. Write $M_{n} (A)$ for the $n \times n$ matrices over $A$ , a C*-algebra, and form the algebraic inductive limit $M_{\infty} (A) = lim M_{n} (A)$ under the corner embeddings $a \mapsto diag (a, 0)$ 39.01.01. Two projections $p, q \in M_{\infty} (A)$ are Murray-von Neumann equivalent, $p \sim q$ , when $p = v^{*} v$ and $q = v v^{*}$ for a partial isometry $v \in M_{\infty} (A)$ ; the equivalence classes form an abelian monoid $V (A)$ under $[p] + [q] = [p \oplus q]$ , where $p \oplus q = diag (p, q)$ . For unital $A$ the $K_{0}$ group is the Grothendieck group $K_{0} (A) = G (V (A))$ 03.08.01. Every element of $K_{0} (A)$ is a formal difference $[p] - [q]$ , and $[p] = [q]$ in $K_{0} (A)$ iff $p \oplus r \sim q \oplus r$ for some projection $r$ (stable equivalence) ^{[Rørdam-Larsen-Laustsen Ch. 2]}.

For non-unital $A$ the naive Grothendieck construction is corrected through the unitisation $A = A \oplus C$ , with the split exact sequence $0 \to A \to A π C \to 0$ . One sets $K_{0} (A) = ker (π_{*} : K_{0} (A) \to K_{0} (C))$ . This standard picture agrees with $G (V (A))$ when $A$ is unital and is functorial in $A$ .

The $K_{1}$ group is built from unitaries. For unital $A$ let $U_{n} (A)$ be the unitary group of $M_{n} (A)$ and $U_{\infty} (A) = lim U_{n} (A)$ under $u \mapsto diag (u, 1)$ . Then $K_{1} (A) = U_{\infty} (A) / U_{\infty} (A)_{0}$ , the quotient by the connected component of the identity, equivalently $π_{0} (U_{\infty} (A))$ . The group operation is induced by $\oplus$ , and it agrees with the pointwise product because $diag (u, v)$ and $diag (uv, 1)$ are connected through the rotation $diag (u, 1) R diag (v, 1) R^{*}$ with $R$ a path of unitaries swapping the two coordinates; hence $K_{1} (A)$ is abelian. For non-unital $A$ one again passes to $A$ . Both constructions are half-exact, homotopy-invariant, stable, and continuous (commute with inductive limits); these four properties, established below, are what make operator K-theory computable.

Counterexamples to common slips

Murray-von Neumann equivalence is not unitary equivalence inside $A$ when $A$ is non-unital or the projections have different "complements." In $M_{2}$ the rank-one projections onto two lines are unitarily equivalent, but in general $p \sim q$ only forces a partial isometry, and $p \oplus 1 \sim q \oplus 1$ (stable equivalence) can hold without $p \sim q$ in $A$ itself. $K_{0}$ records the stable class.
$K_{0} (A) = G (V (A))$ is the correct definition only for unital $A$ . For non-unital $A$ , using $G (V (A))$ directly can give the wrong answer (e.g., it fails to be half-exact); the unitisation-kernel picture is mandatory. A symptom: $V (A)$ may collapse to zero while $K_{0} (A)$ is not.
$K_{1} (A)$ is the component group of stabilised unitaries $U_{\infty} (A)$ , not of $U (A)$ . For $A = C (T)$ the winding number makes $K_{1} = Z$ , but the stabilisation is essential: $U_{1}$ already detects winding here, yet for general $A$ a unitary may only become non-null after enlarging the matrix size, and conversely a noncontractible loop in $U_{1}$ can die in $U_{2}$ .

Key theorem with proof Intermediate+

Theorem (stability of $K$ -theory). Let $A$ be a C-algebra and $K = K (H)$ the algebra of compact operators on a separable infinite-dimensional Hilbert space. The corner embedding $ι : A \to A \otimes K$ , $a \mapsto a \otimes e_{11}$ , induces isomorphisms* $$ \iota_* : K_0(A) \xrightarrow{\ \cong\ } K_0(A \otimes \mathbb{K}), \qquad \iota_* : K_1(A) \xrightarrow{\ \cong\ } K_1(A \otimes \mathbb{K}). $$ In particular K-theory is unchanged by tensoring with matrices: $K_(M_n(A)) \cong K_*(A)$.* ^{[Blackadar Ch. 6]}

Proof. Treat $K_{0}$ ; $K_{1}$ is parallel with unitaries replacing projections. The key structural fact is $A \otimes K = lim M_{n} (A)$ , the inductive limit of matrix amplifications under the upper-left corner embeddings $M_{n} (A) ↪ M_{n + 1} (A)$ . By the continuity property of $K_{0}$ (proved in the Full proof set), $K_{0}$ commutes with inductive limits, so $$ K_0(A \otimes \mathbb{K}) = K_0\big(\varinjlim M_n(A)\big) \cong \varinjlim K_0(M_n(A)). $$ It therefore suffices to show each corner inclusion $M_{n} (A) ↪ M_{n + 1} (A)$ induces an isomorphism on $K_{0}$ compatible with the limit, equivalently that $K_{0} (M_{n} (A)) ≅ K_{0} (A)$ canonically and the limit maps are identities under this identification.

For the matrix-stability isomorphism $K_{0} (M_{n} (A)) ≅ K_{0} (A)$ , observe that $M_{m} (M_{n} (A)) = M_{mn} (A)$ , so the projection monoids satisfy $V (M_{n} (A)) = V (A)$ on the nose: a projection in $M_{m} (M_{n} (A))$ is a projection in $M_{mn} (A)$ , and Murray-von Neumann equivalence is computed in the same ambient $M_{\infty} (A)$ . The embedding $a \mapsto a \otimes e_{11}$ sends a projection $p \in M_{m} (A)$ to $diag (p, 0, \dots, 0) \in M_{mn} (A)$ , which is Murray-von Neumann equivalent to $p$ via the partial isometry collapsing the padding. Hence $ι_{*}$ is the identity on $V$ , an isomorphism after Grothendieck completion.

The corner maps $M_{n} (A) ↪ M_{n + 1} (A)$ likewise act as the identity on $V (A)$ for the same reason: padding by a zero row and column does not change the Murray-von Neumann class. So the inductive system $K_{0} (M_{n} (A))$ is the constant system at $K_{0} (A)$ with identity bonding maps, whose limit is $K_{0} (A)$ , and the composite identification with $K_{0} (A \otimes K)$ is exactly $ι_{*}$ . Therefore $ι_{*} : K_{0} (A) \to K_{0} (A \otimes K)$ is an isomorphism. $□$

Bridge. Stability builds toward the entire computational apparatus of the subject and appears again in 39.02.01, where it is precisely the reason the dimension group of an AF algebra is well-defined independent of the matrix size used to represent a projection: padding by zeros is invisible to $K_{0}$ . The foundational reason the proof works is exactly that $A \otimes K$ is the inductive limit of the matrix amplifications, so this is exactly the continuity property applied to the canonical stabilising system. Stability generalises the matrix-size independence already implicit in the rank invariant of the Beginner worked example, and putting these together with homotopy invariance gives the four pillars — half-exactness, homotopy invariance, stability, continuity — from which the index map and the six-term sequence are assembled; the bridge is that operator K-theory is a stable, homotopy-invariant, half-exact functor, the abstract shape shared with the topological K-theory of 03.08.01.

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Show that $K_{0}$ is a functor: a $*$ -homomorphism $ϕ : A \to B$ induces a group homomorphism $ϕ_{*} : K_{0} (A) \to K_{0} (B)$ , and $(ψ \circ ϕ)_{*} = ψ_{*} \circ ϕ_{*}$ .

Hint

Apply $ϕ$ entrywise to projections in $M_{\infty} (A)$ ; check it respects $\sim$ and $\oplus$ .

Answer

Amplify $ϕ$ to $ϕ_{\infty} : M_{\infty} (A) \to M_{\infty} (B)$ entrywise. A projection $p$ maps to a projection $ϕ_{\infty} (p)$ because $ϕ$ preserves the relations $p^{2} = p = p^{*}$ . If $p = v^{*} v$ , $q = v v^{*}$ then $ϕ_{\infty} (v)$ is a partial isometry implementing $ϕ_{\infty} (p) \sim ϕ_{\infty} (q)$ , so $ϕ$ descends to a monoid map $V (A) \to V (B)$ ; it respects $\oplus$ since $ϕ_{\infty} (diag (p, q)) = diag (ϕ_{\infty} (p), ϕ_{\infty} (q))$ . Grothendieck completion is functorial, giving $ϕ_{*} : K_{0} (A) \to K_{0} (B)$ , and functoriality of completion together with $ϕ_{\infty}$ being a homomorphism gives $(ψ \circ ϕ)_{*} = ψ_{*} \circ ϕ_{*}$ and $(id)_{*} = id$ .

Rubric: full credit for the amplified map preserving projections, $\sim$ , $\oplus$ , and the composition law via functoriality of Grothendieck completion.

Exercise 6 (hard, short-answer).

Prove that $K_{0}$ of an AF algebra $A = lim F_{n}$ equals its dimension group: $K_{0} (A) ≅ lim K_{0} (F_{n})$ as ordered abelian groups, with positive cone the image of $V (A)$ .

Hint

Use continuity of $K_{0}$ and the finite-dimensional computation $K_{0} (F_{n}) = Z^{k_{n}}$ ; projections in an AF algebra perturb into a finite stage.

Answer

By continuity, $K_{0} (lim F_{n}) ≅ lim K_{0} (F_{n})$ . For a finite-dimensional $F_{n} = ⨁_{i} M_{n_{i}}$ one has $K_{0} (F_{n}) = Z^{k_{n}}$ with cone $Z_{\geq 0}^{k_{n}}$ and the connecting maps given by the multiplicity matrices of the Bratteli diagram 39.02.01. The limit ordered group is the dimension group. Positivity passes to the limit because any projection in $M_{\infty} (A)$ is within $ε$ of a self-adjoint element of some $M_{\infty} (F_{n})$ , hence (by functional calculus, $ε$ small) unitarily equivalent to a projection in $F_{n}$ 39.01.01; so $V (A) = lim V (F_{n})$ and the cone is the image of $V (A)$ . Order is preserved because each connecting map is positive. Thus $K_{0} (A) ≅ lim K_{0} (F_{n})$ as ordered groups, the dimension group, recovering the invariant of Elliott's theorem.

Rubric: full credit for continuity, the finite-dimensional cone, projection perturbation into a stage, and the ordered-limit identification.

Exercise 7 (hard, short-answer).

Let $0 \to J \to A π A / J \to 0$ be a short exact sequence of C*-algebras. Define the index map $\partial : K_{1} (A / J) \to K_{0} (J)$ on a unitary lift and prove it is well-defined as a group homomorphism on the class level.

Hint

Lift a unitary $u \in U_{n} (A / J)$ to a partial isometry or, via $diag (u, u^{*})$ , to a unitary $w \in U_{2 n} (A)$ ; then $w diag (1, 0) w^{*}$ and $diag (1, 0)$ differ by an element mapping to a projection over $J$ .

Answer

Given $[u] \in K_{1} (A / J)$ with $u \in U_{n} (A / J)$ , the element $diag (u, u^{*}) \in U_{2 n} (A / J)$ is connected to $1$ , so it lifts to a unitary $w \in U_{2 n} (A)$ with $π_{2 n} (w) = diag (u, u^{*})$ (homotopy lifting along the surjection of unitary groups, using that $U_{0}$ lifts). Set $p = w diag (1_{n}, 0_{n}) w^{*}$ , a projection in $M_{2 n} (A)$ with $π (p) = diag (1_{n}, 0_{n})$ , so $p - diag (1_{n}, 0_{n}) \in M_{2 n} (J)$ (more precisely in $M_{2 n} (J)$ mapping to $0$ in $M_{2 n} (C)$ ). Define $$ \partial[u] = [p] - [\operatorname{diag}(1_n,0_n)] \in K_0(J). $$ Well-definedness: a different unitary lift $w^{'}$ satisfies $w^{'} = w k$ with $k$ a unitary lift of $1$ , hence $k$ homotopic to $1$ in $U_{2 n} (A)$ keeping $π (k) = 1$ , so the resulting projections are homotopic over $J$ and define the same $K_{0} (J)$ class; independence of $n$ and of the representative $u$ in its $K_{1}$ class follows from stabilising and the homotopy invariance of $K_{0}$ . Additivity $\partial ([u] + [v]) = \partial [u] + \partial [v]$ comes from $diag (u, v)$ lifting to $diag (w_{u}, w_{v})$ and $[p_{u} \oplus p_{v}] = [p_{u}] + [p_{v}]$ . Hence $\partial$ is a well-defined homomorphism.

Rubric: full credit for the $diag (u, u^{*})$ unitary lift, the projection difference landing in $K_{0} (J)$ , well-definedness via lift ambiguity being null-homotopic, and additivity.

Lean formalization Intermediate+

lean_status: none — Mathlib has C*-algebras, matrices over them, projection and unitary predicates, and the abstract Grothendieck group of a monoid, but it does not assemble $V (A)$ , $K_{0} (A)$ (with the unitisation-corrected non-unital picture and the order $K_{0} (A)_{+}$ ), or $K_{1} (A) = π_{0} (U_{\infty} (A))$ , nor the functoriality, stability, homotopy-invariance, half-exactness, continuity, the index map, or the six-term exact sequence.

The intended statements read schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.CStarAlgebra.Matrix

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

/-- Stability: tensoring with the compacts does not change K-theory. -/
theorem K0_stable (ι : A →⋆ₐ[ℂ] (A ⊗ Compacts)) (hι : IsCornerEmbedding ι) :
    Function.Bijective (K0.map ι) :=
  sorry  -- A ⊗ 𝕂 = colim Mₙ(A); continuity + padding-invariance of V(A)

/-- Functoriality of K₀. -/
theorem K0_functor (φ : A →⋆ₐ[ℂ] B) :
    ∃ φ₀ : K0 A →+ K0 B, ∀ p : Projection (Matrices A), φ₀ (V.class p) = V.class (φ.amplify p) :=
  sorry

/-- Basic computations. -/
theorem K0_C : K0 ℂ ≃+ ℤ := sorry
theorem K0_BH (H : Type*) [HilbertSpace H] [InfiniteDimensional ℂ H] : K0 (BoundedOp H) ≃+ PUnit := sorry

Advanced results Master

The four pillars and the six-term exact sequence. Operator K-theory is the universal functor on C*-algebras that is homotopy invariant, stable, and half-exact. From half-exactness alone, with the suspension $S A = C_{0} (R) \otimes A$ , one obtains a long exact sequence; Bott periodicity for C*-algebras, $K_{0} (A) ≅ K_{1} (S A) ≅ K_{0} (S^{2} A)$ , collapses it into the six-term exact sequence $$ \begin{array}{ccccc} K_0(J) & \to & K_0(A) & \to & K_0(A/J) \ \uparrow{\scriptstyle\partial_1} & & & & \downarrow{\scriptstyle\partial_0} \ K_1(A/J) & \leftarrow & K_1(A) & \leftarrow & K_1(J) \end{array} $$ attached to any short exact sequence $0 \to J \to A \to A / J \to 0$ , with the exponential map $\partial_{0} : K_{0} (A / J) \to K_{1} (J)$ and the index map $\partial_{1} : K_{1} (A / J) \to K_{0} (J)$ . This is the computational engine: the Pimsner-Voiculescu sequence used in 39.02.01 is the six-term sequence for the crossed-product mapping torus, and the Toeplitz extension $0 \to K \to T \to C (T) \to 0$ has index map $\partial_{1} : K_{1} (C (T)) = Z \to K_{0} (K) = Z$ the Fredholm index, equal to minus the winding number 39.01.04.

The standard computations. The pillars reduce everything to small cases. For the scalars, $V (C) = N$ by rank, so $K_{0} (C) = Z$ and $K_{1} (C) = 0$ since $U_{\infty} (C) = lim U_{n}$ is connected. Matrix stability gives $K_{0} (M_{n}) = Z$ , $K_{1} (M_{n}) = 0$ . For the compacts, $K = lim M_{n}$ with the corner maps, so continuity gives $K_{0} (K) = Z$ , $K_{1} (K) = 0$ — the rank survives in the limit because each corner map is the identity on $K_{0}$ . For $B (H)$ proper infiniteness ( $1 \sim p < 1$ , an Eilenberg swindle) forces $K_{0} (B (H)) = K_{1} (B (H)) = 0$ ; the Calkin algebra $Q (H) = B (H) / K$ then has $K_{0} (Q (H)) = 0$ and $K_{1} (Q (H)) = Z$ , the index of essentially unitary operators, recovering Brown-Douglas-Fillmore. For an AF algebra, continuity gives $K_{0} =$ the dimension group and $K_{1} = 0$ (each $F_{n}$ has connected unitary group). For commutative $A = C (X)$ , the Serre-Swan theorem identifies finitely generated projective $C (X)$ -modules with vector bundles, so $K_{0} (C (X)) = K^{0} (X)$ and $K_{1} (C (X)) = K^{1} (X) = K^{0} (Σ X)$ topological K-theory 03.08.01.

Order and the positive cone. For a stably finite C*-algebra (one in which $1 \oplus p \sim 1$ forces $p = 0$ , e.g. any AF algebra or any C*-algebra with a faithful trace) the image of $V (A)$ in $K_{0} (A)$ is a genuine cone making $(K_{0} (A), K_{0} (A)_{+}, [1_{A}])$ an ordered abelian group with order unit; this is the dimension group when $A$ is AF, and a faithful trace $τ$ induces a state $τ_{*} : K_{0} (A) \to R$ on the ordered group 39.02.01. Proper infiniteness destroys the order: when $[1] = 0$ the cone is everything and the order collapses, as for $B (H)$ , $K$ unitised wrongly, or the Cuntz algebras $O_{n}$ with $K_{0} = Z / (n - 1)$ 39.01.04. The dichotomy — ordered $K_{0}$ for stably finite, torsion or vanishing $K_{0}$ for purely infinite — is the K-theoretic shadow of the type classification.

Synthesis. Operator K-theory is the central insight that organises the entire classification program: it is the foundational reason a noncommutative C*-algebra has a computable topology, because the two functors $K_{0}, K_{1}$ are stable, homotopy invariant, and half-exact, and putting these together with Bott periodicity yields the six-term exact sequence from which every concrete group is read off. This is exactly the noncommutative extension of the topological K-theory of 03.08.01: the Serre-Swan theorem makes $K_{0} (C (X)) = K^{0} (X)$ , so projections in matrix algebras over $C (X)$ are dual to vector bundles on $X$ , and operator K-theory generalises this to algebras with no points at all. The dimension-group invariant of 39.02.01 is exactly $K_{0}$ specialised to AF algebras, where stability is the reason matrix size is invisible and continuity is the reason the limit of dimension groups is the dimension group of the limit; the bridge is that the index map $\partial : K_{1} (A / J) \to K_{0} (J)$ is dual to the connecting map of the extension, turning the Toeplitz, Cuntz, and crossed-product extensions of 39.01.04 into Fredholm-index computations, and the central insight that classifying C*-algebras has become computing an exact sequence of abelian groups is the launch of the Elliott program and of Kasparov's $K K$ -theory.

Full proof set Master

Proposition (continuity: $K_{0}$ commutes with inductive limits). Let $A = lim (A_{n}, ϕ_{n})$ be an inductive limit of C*-algebras with connecting $*$ -homomorphisms $ϕ_{n} : A_{n} \to A_{n + 1}$ . Then $K_{0} (A) ≅ lim (K_{0} (A_{n}), (ϕ_{n})_{*})$ , and likewise for $K_{1}$ .

Proof. A projection $p \in M_{k} (A)$ lies within distance $ε < 1$ of an entry-wise approximant $ϕ_{\infty, n} (a)$ of a self-adjoint $a \in M_{k} (A_{n})$ for some $n$ , because $⋃_{n} ϕ_{\infty, n} (A_{n})$ is dense in $A$ . For $ε$ small the spectrum of $a$ misses $[ε, 1 - ε]$ , so the spectral projection $q = χ_{(1/2, \infty)} (a)$ is a projection in $M_{k} (A_{n})$ (functional calculus, 39.01.01) with $∥ p - ϕ_{\infty, n} (q) ∥ < 1$ , hence $p \sim ϕ_{\infty, n} (q)$ by the standard fact that close projections are unitarily equivalent. Thus every class in $V (A)$ comes from some $V (A_{n})$ , so the natural map $lim V (A_{n}) \to V (A)$ is surjective; injectivity follows because an equivalence $ϕ_{\infty, n} (q) \sim ϕ_{\infty, n} (q^{'})$ in $A$ is implemented by a partial isometry approximable from a finite stage $A_{m}$ , $m \geq n$ , giving $ϕ_{m, n} (q) \sim ϕ_{m, n} (q^{'})$ in $A_{m}$ . Hence $V (A) = lim V (A_{n})$ , and Grothendieck completion commutes with inductive limits of monoids, giving $K_{0} (A) = lim K_{0} (A_{n})$ . The $K_{1}$ statement is identical with unitaries and the fact that a unitary near a unitary is homotopic to it. $□$

Proposition ( $K_{0} (C) = Z$ , $K_{1} (C) = 0$ ). A projection in $M_{n} (C)$ is classified up to Murray-von Neumann equivalence by its rank, so $V (C) = N$ with addition, whose Grothendieck group is $Z$ , generated by the class of a rank-one projection. For $K_{1}$ , the unitary groups $U_{n} (C) = U (n)$ are path connected (every unitary matrix is $exp (i H)$ for self-adjoint $H$ , and $t \mapsto exp (i t H)$ joins it to $1$ ), so $U_{\infty} (C)_{0} = U_{\infty} (C)$ and $K_{1} (C) = 0$ . $□$

Proposition (Eilenberg swindle: $K_{0} (B (H)) = 0$ ). Since $H ≅ H \oplus H$ , there is an isometry $s : H \to H$ with $s s^{*} = p$ a proper projection and $s^{*} s = 1$ , so $1 \sim p$ with $1 - p \neq = 0$ ; iterating, $B (H)$ contains a sequence of mutually orthogonal projections each $\sim 1$ . For any projection $q$ , choosing an infinite orthogonal family $q_{1}, q_{2}, \dots$ with each $q_{i} \sim q$ and $r = \sum q_{i}$ a projection (strong-operator convergence inside $B (H)$ ), one has $[q] + [r] = [r]$ in $V (B (H))$ , so $[q] = 0$ in the Grothendieck group $K_{0} (B (H))$ . As $q$ was arbitrary and projections generate, $K_{0} (B (H)) = 0$ . The same swindle on unitaries (or homotopy invariance: $B (H)$ has a contractible unitary group by Kuiper's theorem) gives $K_{1} (B (H)) = 0$ . $□$

Proposition (matrix stability $K_{0} (M_{n} (A)) ≅ K_{0} (A)$ ). Because $M_{m} (M_{n} (A)) = M_{mn} (A)$ , the monoid $V (M_{n} (A))$ , computed from projections in $lim_{m} M_{m} (M_{n} (A)) = lim_{m} M_{mn} (A) = M_{\infty} (A)$ , equals $V (A)$ . The corner inclusion $A ↪ M_{n} (A)$ , $a \mapsto a \otimes e_{11}$ , sends $[p]$ to the class of $diag (p, 0, \dots, 0)$ , equal to $[p]$ since padding by zeros is a Murray-von Neumann equivalence. Hence the induced map on $V$ is the identity, and $K_{0} (M_{n} (A)) = G (V (M_{n} (A))) = G (V (A)) = K_{0} (A)$ . $□$

Proposition (the index map is exact at $K_{1} (A / J)$ ). For $0 \to J \to A π A / J \to 0$ , the composite $K_{1} (A) π_{*} K_{1} (A / J) \partial K_{0} (J)$ is zero: a unitary $u = π (\tilde{u})$ lifting from $U_{n} (A)$ has $diag (u, u^{*})$ lifting to the unitary $w = diag (\tilde{u}, \tilde{u}^{*}) \in U_{2 n} (A)$ , whence $p = w diag (1, 0) w^{*} = diag (\tilde{u} 1 \tilde{u}^{*}, 0) = diag (1, 0)$ , so $\partial [u] = [p] - [diag (1, 0)] = 0$ . Conversely if $\partial [u] = 0$ then the projection $p$ is connected over $J$ to $diag (1, 0)$ , and the unitary $w$ implementing this can be adjusted by a unitary over $J$ to lie in the image of $U (A)$ , exhibiting $[u] \in im π_{*}$ . This is the exactness used to splice the six-term sequence; the remaining exactness statements are the half-exactness of $K_{0}, K_{1}$ together with the Bott isomorphism $K_{0} (S A) ≅ K_{1} (A)$ . $□$

Connections Master

AF Algebras, Bratteli Diagrams, and the Irrational Rotation Algebra 39.02.01 — the dimension group $(K_{0} (A), K_{0} (A)_{+}, Σ (A))$ classifying AF algebras is exactly $K_{0}$ of this unit equipped with its order and scale; continuity of $K_{0}$ is the reason the dimension group is the limit of the finite-stage $Z^{k_{n}}$ , and the Pimsner-Voiculescu computation of $K_{0} (A_{θ}) ≅ Z^{2}$ is the six-term sequence applied to a crossed product.
C-algebras: axioms, spectrum, and the continuous functional calculus 39.01.01* — projections are produced from self-adjoint elements by the functional calculus, and the perturbation argument that close projections are unitarily equivalent (used throughout the continuity and stability proofs) is functional calculus applied to spectral gaps; the matrix amplifications $M_{n} (A)$ are the C*-algebraic substrate of $M_{\infty} (A)$ .
Topological K-theory 03.08.01 — operator K-theory is the noncommutative extension of the vector-bundle K-theory there: the Serre-Swan theorem makes $K_{0} (C (X)) = K^{0} (X)$ and $K_{1} (C (X)) = K^{1} (X)$ , so projections over $C (X)$ correspond to bundles on $X$ , and the abstract stable/homotopy-invariant/half-exact axioms are shared between the two theories.
Toeplitz and Cuntz Algebras and Extensions 39.01.04 — the six-term exact sequence turns the Toeplitz extension $0 \to K \to T \to C (T) \to 0$ into the Fredholm-index map $\partial : K_{1} (C (T)) = Z \to K_{0} (K) = Z$ , and computes $K_{0} (O_{n}) = Z / (n - 1)$ for the Cuntz algebras, the purely-infinite counterpart to the stably finite dimension groups.
Comparison of Projections and the Murray-von Neumann Type Classification 39.03.04 — Murray-von Neumann equivalence of projections is the same relation refined in the type classification of von Neumann algebras; stable finiteness (ordered $K_{0}$ ) versus proper infiniteness ( $K_{0}$ with $[1] = 0$ ) is the K-theoretic image of the finite/infinite dichotomy of factors.

Historical & philosophical context Master

The transport of Grothendieck's group completion from algebraic geometry into topology was carried out by Atiyah and Hirzebruch in 1959-1961, defining $K^{0} (X)$ from vector bundles and proving the Riemann-Roch theorem in K-theoretic form ^{[Atiyah-Hirzebruch 1961]}. The bridge to operator algebras runs through the Serre-Swan theorem: Swan showed in 1962 that finitely generated projective modules over $C (X)$ are exactly the sections of vector bundles, so the algebraic $K_{0}$ of the ring $C (X)$ coincides with the topological $K^{0} (X)$ ^{[Swan 1962]}. This made it natural to define $K_{0}$ of an arbitrary C*-algebra through projections in matrix amplifications.

The $K_{1}$ group as components of the unitary group, the stability and homotopy-invariance properties, and the six-term exact sequence were developed in the 1970s, with the index map identified with the Fredholm index through the Toeplitz extension and Brown-Douglas-Fillmore theory. Blackadar's monograph consolidated the subject, and Cuntz's picture of $K$ -theory via quasihomomorphisms and Kasparov's $K K$ -theory placed $K_{0}$ and $K_{1}$ inside a bivariant framework in which the index map and Bott periodicity become formal consequences ^{[Blackadar 1998]}. Connes's program of noncommutative geometry took operator K-theory as the homology theory of noncommutative spaces, pairing it with cyclic cohomology through the Chern character.

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.
Wegge-Olsen, N. E., K-Theory and C-Algebras: A Friendly Approach*, Oxford University Press, 1993.
Davidson, K. R., C-Algebras by Example*, Fields Institute Monographs 6, American Mathematical Society, 1996. Ch. IV-VII.
Swan, R. G., "Vector bundles and projective modules", Transactions of the American Mathematical Society 105 (1962), 264-277.
Atiyah, M. F. and Hirzebruch, F., "Riemann-Roch theorems for differentiable manifolds", Bulletin of the American Mathematical Society 65 (1959), 276-281.
Cuntz, J., "K-theory for certain C*-algebras", Annals of Mathematics 113 (1981), 181-197.
Brown, L. G., Douglas, R. G., and Fillmore, P. A., "Extensions of C*-algebras and K-homology", Annals of Mathematics 105 (1977), 265-324.

Operator-algebras spine, second unit of the AF-algebras / K-theory chapter. Produced as the K-theory engine: $K_{0}$ as the Grothendieck group of Murray-von Neumann classes of projections in $M_{\infty} (A)$ , $K_{1}$ as $π_{0} (U_{\infty} (A))$ , the four pillars (functoriality, stability $K_(A\otimes\mathbb{K})=K_*(A) $, h o m o t o p y in v a r ian ce, ha l f / s pl i t e x a c t n ess an d co n t in u i t y), t h es i x - t er m e x a c t se q u e n ce an d in d e x ma p, t h es t an d a r d co m p u t a t i o n s$ K_0(\mathbb{C})=\mathbb{Z} $,$ K_0(B(H))=0 $,$ K_0(M_n)=\mathbb{Z} $, A F d im e n s i o n g r o u p,$ K_0(C(X))=K^0(X)$, and the Serre-Swan bridge to the topological K-theory of 03.08.01.*

Prerequisites

39.02.01
39.01.01

Tier anchors

beginner: Strogatz-style 'count the projections you can build, allow formal differences, and count the loops of unitaries you cannot untwist'; the bundles-become-projections picture of Serre-Swan
intermediate: Rørdam-Larsen-Laustsen *An Introduction to K-Theory for C*-Algebras* Ch. 1-8; Davidson *C*-Algebras by Example* Ch. IV-VI
master: Blackadar *K-Theory for Operator Algebras* (MSRI 5) Ch. 1-9; Davidson Ch. V-VII; Wegge-Olsen *K-Theory and C*-Algebras*

References

Rørdam, M., Larsen, F., Laustsen, N. J. — An Introduction to K-Theory for C*-Algebras · London Math. Soc. Student Texts 49 (2000), Ch. 1-8: K_0 and K_1, functoriality, stability, half-exactness, the index map
Blackadar, B. — K-Theory for Operator Algebras · MSRI Publications 5, 2nd ed. (1998), Ch. 1-9: K_0, K_1, stability K_*(A⊗K)=K_*(A), homotopy invariance, six-term sequence
Davidson, K. R. — C*-Algebras by Example · Fields Institute Monographs 6 (1996), Ch. IV-VII: K-theory of C*-algebras, AF dimension groups, the rotation algebras
Swan, R. G. — Vector bundles and projective modules · Trans. Amer. Math. Soc. 105 (1962), 264-277: projections over C(X) correspond to vector bundles on X

Estimated time

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