39.01.04 · operator-algebras / c-star-algebras-basics

The Toeplitz Algebra, Cuntz Algebras, and Extensions

shipped3 tiersLean: none

Anchor (Master): Davidson *C*-Algebras by Example* Ch. V and Ch. IX; Cuntz, *Simple C*-algebras generated by isometries* (1977); Brown-Douglas-Fillmore (1973, 1977); Blackadar *K-Theory for Operator Algebras*

Intuition Beginner

Imagine an infinite bookshelf with slots numbered $0, 1, 2, \dots$ , each holding one book. The shift is the move that pushes every book one slot to the right: slot $0$ goes to slot $1$ , slot $1$ to slot $2$ , and so on. Slot $0$ is now empty. This move is one-way: you can always shift right, but if you shift back to the left you lose whatever was in slot $0$ , because it had nowhere to come from. The shift preserves all lengths and overlaps, yet it cannot be undone. That single feature — a length-preserving move that still cannot be reversed — is what makes operator algebras richer than ordinary linear algebra, where length-preserving always means reversible.

The algebra built from this one-way shift is the Toeplitz algebra. It contains the shift, its mirror move, and everything you can assemble from them. Hidden inside it are the "small" operators that only touch finitely much of the shelf, and once you ignore those small corrections, the shift behaves like a plain rotation of a circle. So the Toeplitz algebra sits exactly between two familiar worlds: the world of tiny finite operators and the world of functions on a circle.

You can also play this game with several shifts at once, each landing in a different copy of the shelf, packed so they fill everything with no overlap. The algebras you get — the Cuntz algebras — are the purest examples of "infinitely self-similar" operator systems: every piece of them looks like the whole.

Visual Beginner

The shift moves each basis vector to the next one; its mirror moves each back and erases the first.

The dictionary reads: the shift $S$ is the push-down move; its star $S^{*}$ is the pull-up move that drops the first slot; the difference $1 - S S^{*}$ is the small operator that only sees slot $0$ ; and once you discard all such small operators, what is left of $S$ is the circle function that wraps the circle once around itself.

Worked example Beginner

Work on the shelf of square-summable sequences with standard basis $e_{0}, e_{1}, e_{2}, \dots$ . The shift is $S e_{k} = e_{k + 1}$ . Its mirror is $S^{*} e_{k} = e_{k - 1}$ for $k \geq 1$ and $S^{*} e_{0} = 0$ — the first slot has nothing to move up, so it empties.

First check the shift preserves overlaps: $S^{*} S e_{k} = S^{*} e_{k + 1} = e_{k}$ , so $S^{*} S = 1$ , the identity. Pushing down then pulling up returns everything. Now do it the other way: $S S^{*} e_{0} = S 0 = 0$ , while $S S^{*} e_{k} = S e_{k - 1} = e_{k}$ for $k \geq 1$ . So $S S^{*}$ keeps every slot except slot $0$ , which it zeroes.

Therefore $1 - S S^{*}$ is the operator that keeps only slot $0$ and discards the rest. Apply it to a vector: it reads off the slot- $0$ entry and throws away everything else. This operator touches only one slot, so it is "small" — it is the simplest compact operator, a rank-one projection.

What this tells us: $S^{*} S = 1$ but $S S^{*} \neq = 1$ ; the gap between them, $1 - S S^{*}$ , is a single small projection. That one-slot gap is the seed of everything: it is the difference between the one-way shift and a genuine rotation, and it is the smallest piece of the ideal of small operators living inside the Toeplitz algebra.

Check your understanding Beginner

Formal definition Intermediate+

Let $H = ℓ^{2} (N)$ with orthonormal basis $(e_{k})_{k \geq 0}$ . The unilateral shift is the isometry $S \in B (H)$ defined by $S e_{k} = e_{k + 1}$ . It satisfies $S^{*} S = 1$ and $S S^{*} = 1 - p_{0}$ , where $p_{0}$ is the rank-one projection onto $C e_{0}$ 02.11.05. The Toeplitz algebra is the unital C*-subalgebra $$ \mathcal{T} = C^(S) \subseteq B(H) $$ generated by $S$ . Because $1 - SS^ = p_0 $i sco m p a c t an d$ \mathcal{T} $a c t s i r r e d u c ib l y,$ \mathcal{T} $co n t ain s t h e i d e a l$ K = K(H) $o f co m p a c t o p er a t or s; t h e q u o t i e n t$ \mathcal{T}/K $i s g e n er a t e d b y t h e ima g e$ u $o f$ S $, w hi c hi s u ni t a r y ($ u^* u = u u^* = 1 $s in ce$ SS^* \equiv 1 \bmod K$). The Toeplitz extension is the short exact sequence $$ 0 \longrightarrow K \longrightarrow \mathcal{T} \xrightarrow{\ \sigma\ } C(\mathbb{T}) \longrightarrow 0, $$ where the symbol map $σ : T \to C (T) = C^{*} (u)$ sends $S \mapsto z$ , the coordinate function on the unit circle ^{[Davidson Ch. V]}. For $f \in C (T)$ the Toeplitz operator $T_{f} = P M_{f} P$ is compression of multiplication by $f$ to the Hardy space $H^{2} ≅ ℓ^{2} (N)$ ; then $σ (T_{f}) = f$ and $T_{f}$ is Fredholm iff $f$ is nowhere zero, with $index (T_{f}) = - wind (f)$ .

Fix $n \geq 2$ . The Cuntz algebra $O_{n}$ is the universal unital C*-algebra generated by isometries $s_{1}, \dots, s_{n}$ subject to $$ s_i^* s_j = \delta_{ij},1 \quad (1 \le i,j \le n), \qquad \sum_{i=1}^n s_i s_i^* = 1. $$ The first relation says each $s_{i}$ is an isometry with $s_{i}^{*} s_{j} = 0$ for $i \neq = j$ (orthogonal ranges); the second says these ranges partition the identity. Universal C-algebra by generators and relations: given generators $G$ and a set $R$ of $ $- p o l y n o mia l r e l a t i o n s a d mi tt in g a u ni f or mn or mb o u n d a cr oss a l l H i l b er t - s p a cer e p r ese n t a t i o n s, t h e u ni v er s a l C * - a l g e b r a$ C^(G \mid R) $i s t h eco m pl e t i o n o f t h e f r ee$ $- a l g e b r ain t h es u p r e m u mn or m$ |x| = \sup_\pi |\pi(x)| $o v er r e p r ese n t a t i o n s$ \pi $s a t i s f y in g$ R $; f or$ O_n $t h er e l a t i o n s f or ce$ |s_i| = 1 $, so t h es u p r e m u mi s f ini t e . T h e * * C u n t z - K r i e g er a l g e b r a * *$ O_A $o f an$ n \times n $ma t r i x$ A \in {0,1}^{n\times n} $w i t hn oz er or o w r e pl a ces t h eseco n d r e l a t i o nb y$ s_i^* s_i = \sum_j A_{ij}, s_j s_j^ $; f or$ A $t h e a l l - o n es ma t r i x t hi sr eco v er s$ O_n $. T h e in f ini t e v er s i o n$ O_\infty $i s g e n er a t e d b y i so m e t r i es$ (s_i)_{i \ge 1} $w i t h or t h o g o na l r an g es,$ s_i^ s_j = \delta_{ij}1 $, an d$ \sum_{i=1}^k s_i s_i^* \le 1 $f or a l l$ k$ (no upper bound imposed).

A C-algebra extension* of $B$ by $J$ is a short exact sequence $0 \to J \to E q B \to 0$ of C*-algebras. Pushing forward via the quotient $π : M (J) \to M (J) / J = Q (J)$ (the corona algebra, with $M (J)$ the multiplier algebra), the extension is classified by the Busby invariant $τ : B \to Q (J)$ , a $*$ -homomorphism, with $E ≅ {(b, m) \in B \oplus M (J) : τ (b) = π (m)}$ the pullback ^{[Busby 1968]}. When $J = K$ , $M (K) = B (H)$ and $Q (K) = B (H) / K$ is the Calkin algebra. Two extensions are equivalent if their Busby maps are unitarily equivalent in $Q (J)$ ; BDF theory studies $Ext (B)$ , the set of these classes for $J = K$ , which carries an abelian semigroup structure (Brown-Douglas-Fillmore: a group for $B = C (X)$ , $X$ compact metric) ^{[Brown-Douglas-Fillmore 1977]}.

Counterexamples to common slips

An isometry need not be unitary. $S^{*} S = 1$ does not give $S S^{*} = 1$ ; the defect $1 - S S^{*}$ measures the failure. Treating the unilateral shift as invertible (as one would a rotation) loses precisely the index information that gives Toeplitz theory its content.
The relations for $O_{n}$ are not redundant. Imposing only $s_{i}^{*} s_{j} = δ_{ij} 1$ without $\sum s_{i} s_{i}^{*} = 1$ gives the Toeplitz-Cuntz algebra (the extension $E_{n}$ ), which is not simple and surjects onto $O_{n}$ ; the partition-of-unity relation is what kills the ideal generated by $1 - \sum s_{i} s_{i}^{*}$ .
$Ext (X)$ is not classified by the index alone in general. For $X = T$ , $Ext (T) ≅ Z$ via the index of an essentially unitary operator; but for higher-dimensional $X$ the group is $K^{1} (X)$ , and a single integer no longer suffices. Reading every BDF class as "an index" is correct only in dimension one.

Key theorem with proof Intermediate+

Theorem (Coburn; the Toeplitz extension and the index map). Let $S$ be the unilateral shift on $H = ℓ^{2} (N)$ and $\mathcal{T} = C^(S) $. T h e n$ K(H) \subseteq \mathcal{T} $, an d t h esy mb o l ma p$ \sigma : \mathcal{T} \to C(\mathbb{T}) $,$ \sigma(S) = z $, i s a s u r j ec t i v e$ $- h o m o m or p hi s m w i t hk er n e l e x a c tl y$ K(H)$, giving the short exact sequence $$ 0 \longrightarrow K(H) \longrightarrow \mathcal{T} \xrightarrow{\ \sigma\ } C(\mathbb{T}) \longrightarrow 0. $$ Moreover any operator $T \in T$ with invertible symbol $σ (T)$ is Fredholm, and the index gives a surjective homomorphism $index : π_{0} (GL (C (T))) \to Z$ with $index (T) = - wind (σ (T))$ , the negative winding number. ^{[Davidson Ch. V; Murphy Ch. 3.5]}

Proof. Compute $1 - S S^{*} = p_{0}$ , the projection onto $C e_{0}$ , a rank-one operator. The C*-algebra $T$ contains $p_{0} = 1 - S S^{*}$ , hence $S^{j} p_{0} (S^{*})^{k} = e_{j} \otimes \overline{e_{k}}$ for all $j, k \geq 0$ , the matrix units of $B (H)$ . Their closed linear span is $K (H)$ , so $K (H) \subseteq T$ . The quotient $T / K$ is generated by $u = S + K$ , and $u^{*} u = S^{*} S + K = 1$ , $u u^{*} = S S^{*} + K = 1 - p_{0} + K = 1$ , so $u$ is unitary in $T / K$ . Thus $T / K$ is a commutative C*-algebra generated by a single unitary, hence isomorphic to $C (σ (u))$ by the continuous functional calculus 39.01.01; since $σ (u) = T$ (the spectrum of the shift modulo compacts is the full circle), $T / K ≅ C (T)$ with $u \mapsto z$ . The induced quotient map is the symbol $σ$ , with kernel $K$ by construction. Surjectivity is immediate, as $z$ generates $C (T)$ .

For the index, let $T \in T$ have invertible symbol $f = σ (T) \in C (T)$ . Then $T + K$ is invertible in $T / K$ , hence $T$ is Fredholm by Atkinson's theorem (invertibility modulo compacts), so $dim ker T < \infty$ , $dim coker T < \infty$ , and $index (T) = dim ker T - dim coker T$ is defined. The index is locally constant on Fredholm operators and additive under composition. For the model Toeplitz operator $T_{f}$ with $f = z^{m}$ , one has $T_{z^{m}} = S^{m}$ when $m \geq 0$ , with $ker S^{m} = 0$ and $coker S^{m}$ of dimension $m$ (the first $m$ basis vectors are missed), so $index (S^{m}) = - m$ ; for $m < 0$ take adjoints, giving $index (T_{z^{m}}) = - m$ throughout. A nowhere-zero $f \in C (T)$ is homotopic through nowhere-zero functions to $z^{wind (f)}$ — this is the statement that $π_{0}$ of $C (T)^{- 1}$ , equivalently $[T, C^{\times}] = [T, S^{1}] = π_{1} (S^{1}) = Z$ , is detected by the winding number — and homotopy invariance of the index gives $index (T_{f}) = index (T_{z^{wind (f)}}) = - wind (f)$ . Surjectivity onto $Z$ follows since $wind (z^{m}) = m$ is arbitrary. $□$

Bridge. The Toeplitz extension builds toward the whole of extension theory and index theory, and it appears again in 39.04.01, where the boundary map of the six-term K-theory exact sequence sends the unitary $z \in C (T)$ to the class $[1] \in K_{0} (K) = Z$ — the index map is the connecting homomorphism $\partial : K_{1} (C (T)) \to K_{0} (K)$ , and this is exactly the topological content of "a Fredholm operator has an integer index". The foundational reason the sequence is exact is the GNS-and-functional-calculus fact from 39.01.03 and 39.01.01 that $1 - S S^{*}$ generates the compacts while the symbol algebra is the functions on the spectrum of $u$ . This generalises the commutative picture: $C (T)$ is the boundary of the disc, the Hardy space is the boundary values of holomorphic functions, and the Toeplitz operator is the boundary compression. Putting these together, the central insight is that the failure of an isometry to be unitary is a single compact projection whose K-theory class is the index, and the bridge is that this one extension generates $Ext (T) ≅ Z$ , the first and simplest case of BDF theory.

Exercises Intermediate+

Exercise 6 (hard, short-answer).

Prove that in $O_{n}$ the range projections $p_{i} = s_{i} s_{i}^{*}$ are mutually orthogonal and sum to $1$ , and deduce that each $p_{i}$ is Murray-von Neumann equivalent to $1$ .

Hint

Use $s_{i}^{*} s_{j} = δ_{ij} 1$ to compute $p_{i} p_{j}$ , and recall $p \sim q$ iff there is a partial isometry $v$ with $v^{*} v = p$ , $v v^{*} = q$ .

Answer

Compute $p_{i} p_{j} = s_{i} s_{i}^{*} s_{j} s_{j}^{*} = s_{i} (δ_{ij} 1) s_{j}^{*} = δ_{ij} s_{i} s_{j}^{*}$ , and for $i = j$ this is $s_{i} s_{i}^{*} = p_{i}$ (using $s_{i}^{*} s_{i} = 1$ so $p_{i}$ is a projection), while for $i \neq = j$ it is $0$ . Hence the $p_{i}$ are orthogonal projections, and $\sum_{i} p_{i} = \sum_{i} s_{i} s_{i}^{*} = 1$ by the second Cuntz relation. For equivalence, $s_{i}$ is a partial isometry with $s_{i}^{*} s_{i} = 1$ and $s_{i} s_{i}^{*} = p_{i}$ , which is exactly $1 \sim p_{i}$ in the Murray-von Neumann sense. So the identity is equivalent to each of $n$ mutually orthogonal subprojections summing to it: $1 \sim p_{1} \sim \dots \sim p_{n}$ with $1 = \sum p_{i}$ . This forces $[1] = n [1]$ in $K_{0}$ , i.e. $(n - 1) [1] = 0$ , the seed of $K_{0} (O_{n}) = Z / (n - 1)$ .

Rubric: full credit for the orthogonality/partition computation and the partial-isometry equivalence $1 \sim p_{i}$ .

Exercise 7 (hard, short-answer).

State the Busby invariant of an extension $0 \to K \to E \to B \to 0$ and explain how it recovers $E$ .

Hint

Push the multiplier algebra $M (K) = B (H)$ to the Calkin algebra $Q (K) = B (H) / K$ and ask where $B$ lands.

Answer

Given the extension, each $b \in B$ lifts to some $e \in E \subseteq M (K) = B (H)$ , well-defined modulo $K$ , so the assignment $τ (b) = e + K \in Q (K) = B (H) / K$ is a $*$ -homomorphism $τ : B \to Q (K)$ , the Busby invariant. It is independent of the lift because two lifts differ by an element of $K = ker (π)$ . To recover $E$ , form the pullback $$ E ;\cong; {(b, m) \in B \oplus B(H) : \tau(b) = \pi(m)}, $$ where $π : B (H) \to Q (K)$ is the Calkin quotient; the projection $(b, m) \mapsto b$ is the surjection onto $B$ with kernel ${(0, m) : π (m) = 0} = K$ . Thus the extension is determined up to isomorphism by $τ$ , and conversely every $*$ -homomorphism $B \to Q (K)$ arises this way, so extensions of $B$ by $K$ are classified by Busby maps. BDF equivalence then quotients by unitary equivalence in $Q (K)$ .

Rubric: full credit for defining $τ$ via the Calkin quotient and exhibiting the pullback reconstruction of $E$ .

Lean formalization Intermediate+

lean_status: none — Mathlib has CStarAlgebra, the compact operators as a closed ideal of B(H), and the continuous functional calculus, but neither the Toeplitz algebra of the unilateral shift, the Toeplitz extension as a named short exact sequence with its symbol map, the universal C*-algebra by generators and relations, the Cuntz algebras $O_{n}$ with simplicity and pure infiniteness, nor the Calkin algebra packaged with the Busby invariant and the BDF group $Ext (X)$ .

The intended statement reads schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.InnerProductSpace.l2Space

open scoped ComplexOrder

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]

/-- The Toeplitz extension: the C*-algebra of the unilateral shift sits in a
short exact sequence with kernel the compacts and quotient C(𝕋). -/
theorem toeplitz_extension
    (S : H →L[ℂ] H) (hiso : (star S) * S = 1) (hdef : 1 - S * (star S) = p₀) :
    ∃ (σ : (StarSubalgebra.adjoin ℂ {S}) →⋆ₐ[ℂ] C(Circle, ℂ)),
      Function.Surjective σ ∧
      RingHom.ker σ = compactOperatorIdeal :=
  sorry  -- 1 - SS* generates K(H); quotient is C*(unitary) ≅ C(𝕋)

Advanced results Master

The two families — Toeplitz/extensions and Cuntz — are the simple end and the index-theoretic end of the same circle of ideas, and the structure theorems below organise both.

Coburn's universal property of $T$ . The Toeplitz algebra is the universal C*-algebra generated by a single isometry: $T = C^{*} (v ∣ v^{*} v = 1)$ . Coburn's theorem states that any nonunitary isometry $V$ on any Hilbert space generates a copy of $T$ via $S \mapsto V$ , and the resulting representation is faithful. Consequently the Wold decomposition — every isometry is a direct sum of a unitary and copies of the shift — is the representation theory of $T$ : the unitary part factors through the symbol $C (T)$ , and the shift parts are the faithful summand. The defect projection $1 - V V^{*}$ is the obstruction to unitarity and carries the index.

Cuntz's theorem: $O_{n}$ is simple and purely infinite. For $2 \leq n \leq \infty$ , $O_{n}$ is simple (no closed two-sided ideals but $0$ and itself), separable, nuclear, and purely infinite (every nonzero hereditary subalgebra contains an infinite projection; equivalently, for every nonzero positive $a$ there are $x, y$ with $x a y = 1$ ). Simplicity follows from the gauge action of $T$ — $λ \cdot s_{i} = λ s_{i}$ — whose fixed-point algebra is the UHF algebra $M_{n^{\infty}} = ⨂ M_{n}$ , together with a conditional expectation onto it; any nonzero ideal, being gauge-invariant after averaging, would meet the simple core. Pure infiniteness comes directly from the relations: words $s_{μ} = s_{μ_{1}} \dots s_{μ_{k}}$ in multi-indices give $s_{μ}^{*} s_{μ} = 1$ while $s_{μ} s_{μ}^{*}$ are arbitrarily small orthogonal projections summing into $1$ , so $1$ is equivalent to a proper subprojection of itself — the hallmark of an infinite algebra. The K-theory is $K_{0} (O_{n}) = Z / (n - 1) Z$ generated by $[1]$ (with $(n - 1) [1] = 0$ forced by $1 \sim n \cdot 1$ ), $K_{1} (O_{n}) = 0$ ; for $O_{\infty}$ , $K_{0} = Z$ and $K_{1} = 0$ . These were the first separable C*-algebras shown to be non-stably-finite, and they anchor the Kirchberg-Phillips classification of purely infinite nuclear simple C*-algebras by K-theory.

Cuntz-Krieger algebras and symbolic dynamics. For a $0$ - $1$ matrix $A$ with no zero rows or columns, $O_{A}$ is simple iff $A$ is irreducible and not a permutation matrix, purely infinite in the same cases, and $K_{0} (O_{A}) = coker (1 - A^{t})$ , $K_{1} (O_{A}) = ker (1 - A^{t})$ on $Z^{n}$ . The algebra $O_{A}$ is the C*-algebra of the subshift of finite type with transition matrix $A$ : its K-theory is the Bowen-Franks group of the shift, making flow-equivalence of subshifts visible inside operator K-theory. This is the prototype of the graph-algebra and groupoid-algebra programs.

BDF theory and $Ext$ . For $X$ a compact metrizable space, an extension $0 \to K \to E \to C (X) \to 0$ is given by its Busby map $τ : C (X) \to Q (H)$ , a unital $*$ -monomorphism (an essentially normal family when $X \subseteq C$ , where $τ$ is generated by an essentially normal operator $T$ , $[T^{*}, T] \in K$ ). Brown, Douglas, and Fillmore proved $Ext (X)$ is an abelian group under direct sum (the inverse exists because $Q (H)$ absorbs $K$ -stably), and that it is naturally isomorphic to $K^{1} (X)$ , the odd topological K-homology of $X$ . For $X \subseteq C$ this says an essentially normal operator is determined up to unitary equivalence modulo compacts by its essential spectrum $σ_{ess} (T)$ together with the index data $index (T - λ)$ for $λ$ in each bounded component of $C ∖ σ_{ess} (T)$ . The unilateral shift, with essential spectrum $T$ and index $- 1$ in the disc, generates $Ext (T) ≅ Z$ ; the normal operators are exactly the split extensions, those with all indices zero.

Synthesis. The defect projection $1 - S S^{*}$ is the foundational reason the entire theory exists: it is a single compact operator, and its K-theory class is the index, so the Toeplitz extension $0 \to K \to T \to C (T) \to 0$ is exactly the geometric realisation of the index map $\partial : K_{1} (C (T)) \to K_{0} (K) = Z$ . This is dual to the Cuntz construction, where instead of one isometry with a compact defect one takes $n$ isometries with no defect at all ( $\sum s_{i} s_{i}^{*} = 1$ ); the central insight is that removing the defect removes the compacts and produces a simple purely infinite algebra rather than an extension, so $T$ and $O_{n}$ are the two ways an algebra of isometries can close up. Putting these together, the Busby invariant generalises the symbol map: the symbol $σ : T \to C (T)$ is the Busby map of the Toeplitz extension, and BDF theory is the statement that all such maps into the Calkin algebra form the K-homology group $K^{1} (X)$ , the dual theory to the K-theory of 39.04.01. The bridge to the rest of operator algebras is that this one defect projection generalises in two directions — to the index theory of elliptic operators (Toeplitz operators are the boundary model) and to the classification program (Cuntz algebras are the purely infinite building blocks), and these reunite in Kasparov's $K K$ -theory, where $Ext$ and $K$ are the two halves of a single bivariant functor.

Full proof set Master

Proposition ($S^ S = 1 $,$ SS^ = 1 - p_0 $, an d$ \sigma(S) = \overline{\mathbb{D}} $,$ \sigma_{\mathrm{ess}}(S) = \mathbb{T} $) . * * O n$ \ell^2(\mathbb{N}) $,$ S^S e_k = S^ e_{k+1} = e_k $, so$ S^S = 1 $. A n d$ SS^ e_0 = 0 $,$ SS^* e_k = e_k $f or$ k\ge 1 $, so$ SS^* = 1 - p_0 $w i t h$ p_0 = e_0 \otimes \overline{e_0} $r ank o n e . F or t h es p ec t r u m :$ S $i s ani so m e t r y, so$ |S| = 1 $an d$ \sigma(S) \subseteq \overline{\mathbb{D}} $; f or$ |\lambda| < 1 $t h e v ec t or$ \xi_\lambda = \sum_k \overline{\lambda}^k e_k \in \ell^2 $s a t i s f i es$ S^* \xi_\lambda = \overline{\lambda},\xi_\lambda $, so$ \overline\lambda \in \sigma_p(S^*) $, w h e n ce$ \lambda \in \sigma(S) $; t h u s t h eo p e n d i sc l i es in$ \sigma(S) $, an d b y c l ose d n ess$ \sigma(S) = \overline{\mathbb{D}} $. M o d u l oco m p a c t s$ S $ma p s t o a u ni t a r y$ u $, so$ \sigma_{\mathrm{ess}}(S) = \sigma(u) \subseteq \mathbb{T} $, an d e q u a l i t y h o l d ss in ce$ u $i s a g e n er a t or o f$ C(\mathbb{T})$ with full spectrum.

Proposition (Atkinson: invertibility mod $K$ equals Fredholm). An operator $T \in B (H)$ is Fredholm iff $T + K$ is invertible in the Calkin algebra $Q (H)$ . If $T$ is Fredholm, a parametrix $R$ with $T R - 1, R T - 1 \in K$ exists (invert $T$ on the orthogonal complement of the finite-dimensional kernel/cokernel and extend by $0$ ), so $T + K$ is invertible. Conversely, if $R T = 1 - k_{1}$ , $T R = 1 - k_{2}$ with $k_{i} \in K$ , then $ker T \subseteq ker (1 - k_{1}) = ker$ of a compact perturbation of the identity, which is finite-dimensional (Riesz theory of compact operators 02.11.05); similarly $coker T$ is finite-dimensional. Hence $T$ is Fredholm.

Proposition ( $index (T_{z^{m}}) = - m$ and homotopy invariance). For $m \geq 0$ , $T_{z^{m}} = S^{m}$ on $H^{2}$ , with $ker S^{m} = 0$ and $coker S^{m} = span {e_{0}, \dots, e_{m - 1}}$ , so $index = - m$ . The index is a continuous, hence locally constant, $Z$ -valued function on the open set of Fredholm operators (a Fredholm operator stays Fredholm under small perturbations with constant index, by the parametrix and Riesz theory). A nowhere-zero $f \in C (T)$ is homotopic within $C (T)^{- 1}$ to $z^{wind (f)}$ because $f /∣ f ∣ : T \to T$ has a continuous logarithm after dividing by $z^{wind (f)}$ (the corrected map is null-homotopic). Lifting the homotopy to Toeplitz operators with continuous symbol keeps the operators Fredholm, so $index (T_{f}) = index (T_{z^{wind (f)}}) = - wind (f)$ .

Proposition (universal C-algebra by generators and relations exists when the relations are admissible).* Let $G$ be a set and $R$ a set of relations of the form $∥ p (g_{1}, \dots, g_{k}) ∥ \leq c$ for $*$ -polynomials $p$ . Call $R$ admissible if for every generator $g$ the quantity $sup {∥ π (g) ∥ : π a * -representation satisfying R}$ is finite. Then on the free unital $*$ -algebra $F$ on $G$ , $∥ x ∥_{u} = sup_{π} ∥ π (x) ∥$ is a C*-seminorm; its kernel $N = {x : ∥ x ∥_{u} = 0}$ is a two-sided $*$ -ideal, and the completion of $F / N$ in $∥ \cdot ∥_{u}$ is a C*-algebra $C^{*} (G ∣ R)$ with the universal property that every representation satisfying $R$ factors through it. For $O_{n}$ the relations force $∥ s_{i} ∥ = 1$ (from $s_{i}^{*} s_{i} = 1$ and $∥ s_{i} ∥^{2} = ∥ s_{i}^{*} s_{i} ∥$ ), so admissibility holds and $O_{n}$ exists; a concrete faithful representation on full Fock space shows it is nonzero.

Proposition ( $O_{n}$ is simple). Let $β : T \to Aut (O_{n})$ be the gauge action $β_{λ} (s_{i}) = λ s_{i}$ , with fixed-point algebra $F = O_{n}^{β}$ and faithful conditional expectation $E (x) = \int_{T} β_{λ} (x) d λ$ . The algebra $F$ is the closure of the span of words $s_{μ} s_{ν}^{*}$ with $∣ μ ∣ = ∣ ν ∣$ , and is isomorphic to the UHF algebra $M_{n^{\infty}}$ , which is simple. Let $J \neq = 0$ be a closed two-sided ideal. For $0 \neq = a \in J$ positive, $E (a) \in J \cap F$ is nonzero (faithfulness of $E$ on positives), and by approximating with finite words one finds elements $x, y \in O_{n}$ with $x E (a) y$ close to a nonzero scalar multiple of a projection $s_{μ} s_{μ}^{*}$ , which is equivalent to $1$ ; hence $1 \in J$ and $J = O_{n}$ . Therefore the only closed two-sided ideals of $O_{n}$ are $0$ and $O_{n}$ itself.

Proposition ( $Ext (T) ≅ Z$ , computed by the index). An extension of $C (T)$ by $K$ is a unital $*$ -monomorphism $τ : C (T) \to Q (H)$ ; equivalently the choice of an essentially unitary $T \in B (H)$ (so $T^{*} T - 1, T T^{*} - 1 \in K$ ) with $τ (z) = T + K$ . Two such are equivalent iff the operators are unitarily equivalent mod $K$ . By Weyl-von Neumann-Berg, an essentially unitary operator is determined mod $K$ up to unitary equivalence by its essential spectrum (here $T$ ) and the index $index (T) \in Z$ . The sum of extensions adds indices, the split extension has index $0$ , and the inverse of index $m$ has index $- m$ , so $Ext (T) \to Z$ , $τ \mapsto index (T)$ , is a group isomorphism. The unilateral shift realises the generator $- 1$ .

Connections Master

C-algebras: axioms, spectrum, and the continuous functional calculus 39.01.01* — the quotient $T / K$ is a commutative C*-algebra generated by one unitary, so the identification $T / K ≅ C (T)$ is the continuous functional calculus of that unitary; spectral permanence and the spectral mapping theorem from that unit fix $σ (u) = T$ .
States, the GNS construction, and Gelfand-Naimark 39.01.03 — the universal C*-algebra on generators and relations is defined by a supremum over representations, each built by GNS from states; that $O_{n}$ and $T$ have faithful representations (on Fock space, on $ℓ^{2} (N)$ ) is what makes the universal norm a norm rather than a seminorm.
Compact operators 02.11.05 — the ideal in the Toeplitz extension is exactly $K (H)$ ; the Riesz theory of compact operators is what makes operators with invertible symbol Fredholm, and the rank-one defect $1 - S S^{*}$ is the generator of the compacts inside $T$ .
K-theory of C-algebras and the six-term sequence 39.04.01* — the index map is the connecting homomorphism $\partial : K_{1} (C (T)) \to K_{0} (K)$ of the Toeplitz extension, and $K_{0} (O_{n}) = Z / (n - 1)$ is computed from the Cuntz relations; BDF $Ext$ is the K-homology dual paired with this K-theory.
Von Neumann algebras and the bicommutant theorem 39.03.01 — the gauge-fixed core of $O_{n}$ is the UHF algebra $M_{n^{\infty}}$ , whose weak closure in a tracial representation is the hyperfinite II $_{1}$ factor; the passage from the simple C*-algebra to its von Neumann completion is the bridge between this unit and factor theory.

Historical & philosophical context Master

The unilateral shift and its algebra were studied by Toeplitz, Hartman-Wintner, and others in the analysis of Toeplitz forms; Coburn established in 1967 the universal property and the exactness of the symbol sequence that bears the algebra's name ^{[Coburn 1967]}. The index of a Toeplitz operator as minus the winding number of its symbol is the one-dimensional case of the Atiyah-Singer index theorem, and historically it is the example that made the analytic index visible as a topological invariant. Cuntz introduced the algebras $O_{n}$ in 1977, giving the first separable simple C*-algebras containing infinite projections and computing their K-theory as $Z / (n - 1)$ ^{[Cuntz 1977]}; the construction by isometries with orthogonal ranges summing to the identity made "universal C*-algebra by generators and relations" a standard tool, and Cuntz-Krieger algebras (1980) tied these to symbolic dynamics.

The extension-theoretic strand began with Busby's 1968 classification of extensions by a $*$ -homomorphism into the corona algebra ^{[Busby 1968]}, and culminated in the work of Brown, Douglas, and Fillmore (1973, 1977), who classified essentially normal operators — those normal modulo compacts — by their essential spectrum and index data, and showed that the set $Ext (X)$ of such extensions is an abelian group naturally isomorphic to topological K-homology $K^{1} (X)$ . Their proof that $Ext$ is a group, and not merely a semigroup, used a deep absorption argument and brought algebraic topology directly into operator theory. The BDF group and the Toeplitz index are the two boundary values of Kasparov's later bivariant $K K$ -theory, in which $Ext (A, B)$ and $K (A, B)$ become the two faces of one functor.

Bibliography Master

Davidson, K. R., C-Algebras by Example*, Fields Institute Monographs 6, American Mathematical Society, 1996. Ch. V, Ch. IX.
Cuntz, J., "Simple C*-algebras generated by isometries", Communications in Mathematical Physics 57 (1977), 173–185.
Cuntz, J. and Krieger, W., "A class of C*-algebras and topological Markov chains", Inventiones Mathematicae 56 (1980), 251–268.
Coburn, L. A., "The C*-algebra generated by an isometry", Bulletin of the American Mathematical Society 73 (1967), 722–726.
Busby, R. C., "Double centralizers and extensions of C*-algebras", Transactions of the American Mathematical Society 132 (1968), 79–99.
Brown, L. G., Douglas, R. G. and Fillmore, P. A., "Extensions of C*-algebras and K-homology", Annals of Mathematics 105 (1977), 265–324.
Murphy, G. J., C-Algebras and Operator Theory*, Academic Press, 1990. Ch. 3.5, Ch. 6.
Blackadar, B., K-Theory for Operator Algebras, 2nd ed., MSRI Publications 5, Cambridge University Press, 1998.

Operator-algebras spine, the index-theory and universal-construction unit of the C-basics chapter. Produced as the Toeplitz/Cuntz/extensions anchor: the unilateral shift and its extension by the compacts, the symbol map and Fredholm index, universal C*-algebras by generators and relations, the simple purely infinite Cuntz and Cuntz-Krieger algebras, the Busby invariant, and BDF / Ext theory in the Calkin algebra.*

Prerequisites

39.01.03
39.01.01
02.11.05

Tier anchors

beginner: Strogatz-style 'a one-way shift that cannot be undone, and the gap it leaves behind'; the bookshelf-shift picture for the unilateral shift
intermediate: Davidson *C*-Algebras by Example* Ch. V (the Toeplitz algebra, Cuntz algebras, extensions); Murphy *C*-Algebras and Operator Theory* Ch. 3.5 and Ch. 6
master: Davidson *C*-Algebras by Example* Ch. V and Ch. IX; Cuntz, *Simple C*-algebras generated by isometries* (1977); Brown-Douglas-Fillmore (1973, 1977); Blackadar *K-Theory for Operator Algebras*

References

Davidson, K. R. — C*-Algebras by Example · Ch. V, the Toeplitz algebra and the Toeplitz extension; Cuntz algebras O_n; Ch. IX, extensions and BDF theory
Cuntz, J. — Simple C*-algebras generated by isometries · Comm. Math. Phys. 57 (1977), 173–185 — the algebras O_n, simplicity, pure infiniteness
Brown, L. G., Douglas, R. G. and Fillmore, P. A. — Extensions of C*-algebras and K-homology · Ann. of Math. 105 (1977), 265–324 — Ext(X) as a group, essentially normal operators classified by index data
Busby, R. C. — Double centralizers and extensions of C*-algebras · Trans. Amer. Math. Soc. 132 (1968), 79–99 — the Busby invariant classifying extensions

Estimated time

beginner: 18m
intermediate: 52m
master: 92m