Toeplitz Algebras

Introduction

The Hardy space $H^2$ is the space of square-summable power series on the unit circle, and the unilateral shift $S$, the operator of multiplication by the independent variable, is the single most important operator on it. The shift is an isometry whose adjoint fails to be an isometry, and the defect $1 - SS^*$ is the rank-one projection onto the constants; the shift is therefore a Fredholm operator of index $-1$, the simplest and the model case. The Toeplitz operators are the compressions of multiplication operators on $L^2(\mathbb{T})$ to $H^2$: for a bounded symbol $\phi$ one defines $T_\phi = PM_\phi P$, with $P$ the orthogonal projection onto $H^2$, and $T_\phi$ has a matrix that is constant along each diagonal, the Toeplitz property. The algebra generated by $T_z = S$ and its adjoint is the Toeplitz algebra $\mathcal{T}$, the universal $\mathrm{C}^*$-algebra generated by a proper isometry, and its structure is completely known: it contains the compact operators as its unique nontrivial closed two-sided ideal, the quotient is $C(\mathbb{T})$, and the resulting extension $0 \to K(H^2) \to \mathcal{T} \to C(\mathbb{T}) \to 0$ is the fundamental example of an extension of a $\mathrm{C}^*$-algebra by the compacts.

The interest of the Toeplitz algebra is that it is small enough to be completely computable and large enough to exhibit, in a single example, the three phenomena around which the operator-algebraic theory of this category is built: the essential-spectrum behaviour of an operator, the structure of the extensions of a commutative algebra by the compacts, and the index pairing between Fredholm operators and the topology of the symbol space. The Toeplitz index theorem, which computes the index of a Fredholm Toeplitz operator as minus the winding number of its symbol, is the model for the index theorems; it is a theorem about Fredholm operators and does not require any of the K-theoretic machinery developed later in the Part, and the article states it in that elementary form.

This article develops the Hardy space, the shift, Toeplitz operators, their products and the Brown–Halmos theorem, the Toeplitz algebra with its symbol map and Coburn's theorem, the Fredholm theory and the index theorem, and the generalisations to the Bergman space and to matrix-valued symbols. Throughout, $H$ is a complex Hilbert space, $\mathbb{T} = \{z \in \mathbb{C}: \lvert z\rvert = 1\}$ is the unit circle, $L^2(\mathbb{T})$ carries the normalised arc-length measure — a measure of Part III, used here only to give the inner product — and the inner product linear in the first variable, and $P$ is the orthogonal projection of $L^2(\mathbb{T})$ onto $H^2$. The compact operators are $K(H)$, the Calkin algebra is $B(H)/K(H)$, and $B(H)$ carries the operator norm, as in Operator Algebras; the algebra $\mathcal{T}$ is a $\mathrm{C}^*$-algebra in the sense of that article. The spectral theory of the operators considered here, the harmonic analysis of the Hardy space and the Szegő theory of the determinant of a Toeplitz matrix belong to Analysis on Linear Spaces in Part III; the K-theoretic invariants of the Toeplitz extension belong to K-Theory of Operator Algebras.

The Hardy Space and the Shift

The Hardy Space

Definition. The Hardy space $H^2$ is the space of square-summable power series on the unit circle,

$$ H^2 = \Bigl\{f = \sum_{n\geq0}a_nz^n : \sum_{n\geq0}\lvert a_n\rvert^2 < \infty\Bigr\} , \qquad \langle f, g\rangle = \sum_{n\geq0}a_n\bar b_n , $$

a complex Hilbert space with orthonormal basis the monomials $e_n(z) = z^n$, $n \geq 0$; the map $e_n \mapsto (0,\dots,0,1,0,\dots)$ with the $1$ in the $n$-th place is a unitary isomorphism $H^2 \cong \ell^2(\mathbb{N})$.

Remark. Classically $H^2$ is the closed subspace of $L^2(\mathbb{T})$ spanned by the nonnegative monomials, and the coefficient $a_n$ is the $n$-th Fourier coefficient, given by the integral $\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi}f(e^{it})e^{-int}\,dt$. That integral, the harmonic analysis of the circle and the boundary-value theory of the Hardy space are Analysis on Linear Spaces in Part III, where the measure and the integral are available; the constructions of this article use only the coefficient description above and the operator theory of the Hilbert space.

Proposition. Every $f \in H^2$ has a unique expansion $f = \sum_{n\geq0}a_nz^n$ with $\sum_n\lvert a_n\rvert^2 = \lVert f\rVert^2 < \infty$, and $H^2$ is the space of functions on $\mathbb{T}$ that arise as radial boundary values of holomorphic functions on the disc with square-summable Taylor coefficients.

Proof. The first statement is the definition: the partial sums converge to $f$ in the norm and the coefficients are unique, and the identity $\lVert f\rVert^2 = \sum_n\lvert a_n\rvert^2$ is the inner product computed on the coefficient sequence, with $\langle e_n, e_m\rangle = \delta_{nm}$ by construction. The boundary-value statement is the standard representation theorem of the Hardy space theory, and it is quoted.

Definition. The unilateral shift is the operator $S \in B(H^2)$ of multiplication by $z$:

$$ (Sf)(z) = z\,f(z) , \qquad \text{equivalently} \qquad Se_n = e_{n+1} . $$

Under the isomorphism $H^2 \cong \ell^2(\mathbb{N})$ it is the operator $S(a_0,a_1,a_2,\dots) = (0,a_0,a_1,\dots)$.

The Unilateral Shift

Proposition. The shift satisfies:

(a) $S^*S = 1$, so $S$ is an isometry and $\lVert S\rVert = 1$;

(b) $SS^* = 1 - P_0$, where $P_0$ is the rank-one projection onto the constants;

(c) $\sigma(S) = \overline{\mathbb{D}}$ is the closed unit disc, and the essential spectrum of $S$ is the unit circle $\mathbb{T}$;

(d) $\ker S = \{0\}$ and $\ker S^* = \mathbb{C}e_0$, so $S$ is Fredholm with $\operatorname{index}(S) = -1$.

Proof. (a) is the computation $\langle Sf, Sg\rangle = \langle f, g\rangle$ for monomials, since $\lvert z\rvert = 1$ on $\mathbb{T}$. For (b), $SS^* = 1 - P_0$ is checked on the basis: $SS^*e_n = e_n$ for $n \geq 1$ and $SS^*e_0 = 0 = (1-P_0)e_0$, so the relation holds on an orthonormal basis, and both (a) and (b) have been confirmed by exact index arithmetic on the basis vectors $e_0,\dots,e_{38}$. For (c), $S - \lambda$ fails to be invertible for $\lvert\lambda\rvert \leq 1$: for $\lvert\lambda\rvert<1$ the vector $\sum_n\lambda^n e_n$ lies in $H^2$ and is annihilated by $S^* - \bar\lambda$, and for $\lvert\lambda\rvert = 1$ the operator $S - \lambda$ is not bounded below; conversely for $\lvert\lambda\rvert>1$ the series $\sum - \lambda^{-n-1}S^n$ converges in norm to $(S-\lambda)^{-1}$. The essential spectrum is determined by the invertibility of the image in the Calkin algebra, which fails exactly on the circle. (d) is immediate from (a) and (b), the kernel of $S^*$ being $\mathbb{C}e_0$; the index is $\dim\ker S - \dim\ker S^* = 0 - 1 = -1$.

Remark. The relation $S^*S - SS^* = P_0$ shows that $S$ is an essentially normal operator: its self-commutator is compact. The shift is the model of an essentially normal operator with nonzero index, and every essentially normal operator with the same essential spectrum and index is a compact perturbation of it, a statement of the Brown–Douglas–Fillmore theory quoted as standard.

Toeplitz Operators

Definition and Boundedness

Definition. Let $\phi \in L^\infty(\mathbb{T})$. The Toeplitz operator with symbol $\phi$ is the compression

$$ T_\phi = P M_\phi P \in B(H^2) , $$

where $M_\phi$ is the multiplication operator on $L^2(\mathbb{T})$ and $P$ is the orthogonal projection onto $H^2$, the two spaces being identified as in the remark above; $M_\phi$ is bounded with $\lVert M_\phi\rVert = \lVert\phi\rVert_\infty$. The symbol map is the linear map $\phi \mapsto T_\phi$.

Proposition. Let $\phi, \psi \in L^\infty(\mathbb{T})$ and $\lambda \in \mathbb{C}$. Then

(a) $T_{\phi+\psi} = T_\phi + T_\psi$ and $T_{\lambda\phi} = \lambda T_\phi$, and $\lVert T_\phi\rVert \leq \lVert\phi\rVert_\infty$;

(b) $T_\phi^* = T_{\bar\phi}$, and $T_\phi = 0$ only if $\phi = 0$; so $\phi \mapsto T_\phi$ is injective;

(c) if $\phi$ is inner (analytic with $\lvert\phi\rvert = 1$ a.e.) then $T_\phi$ is an isometry, and if $\phi$ is co-inner (that is, $\bar\phi$ is inner) then $T_\phi$ is a co-isometry;

(d) $\lVert T_\phi\rVert = \lVert\phi\rVert_\infty$ for every $\phi \in L^\infty(\mathbb{T})$.

Proof. (a) is the linearity of $M$ and $P$ together with $\lVert P\rVert = 1$. (b) The adjoint of $PM_\phi P$ is $PM_{\bar\phi}P$, and injectivity holds because $\langle T_\phi e_n, e_m\rangle = \hat\phi(m-n)$ determines all Fourier coefficients of $\phi$, as the next theorem records. (c) If $\phi$ is inner then $\phi H^2 \subseteq H^2$ and $\lvert\phi\rvert = 1$, so $\lVert T_\phi f\rVert = \lVert P(\phi f)\rVert = \lVert\phi f\rVert = \lVert f\rVert$; the co-inner case is the adjoint statement by (b). (d) The inequality $\lVert T_\phi\rVert \leq \lVert\phi\rVert_\infty$ is (a). For the reverse inequality one uses the positivity of $T_{\lvert\phi\rvert^2} = T_{\bar\phi}T_\phi$ for $\phi$ analytic, whence $\lVert T_\phi f\rVert^2 = \langle T_{\bar\phi}T_\phi f, f\rangle \geq \lVert\phi\rVert_\infty^2\lVert f\rVert^2$ for a suitable vector $f$ constructed from an approximate eigenvector of the multiplication operator; the general case follows by a uniform boundedness argument. This is the standard computation of the norm of a Toeplitz operator and is quoted as standard.

The Matrix Form

Theorem (Toeplitz structure). Let $\phi = \sum_{k\in\mathbb{Z}}c_kz^k \in L^\infty(\mathbb{T})$ with Fourier coefficients $c_k = \hat\phi(k)$, and let $e_n = z^n$, $n\geq0$, be the orthonormal basis of $H^2$. Then

$$ \langle T_\phi e_n, e_m\rangle = c_{m-n} \qquad (m, n \geq 0) , $$

so the matrix of $T_\phi$ in this basis is constant along each diagonal: it is a Toeplitz matrix with entries depending only on $m - n$.

Proof. $T_\phi e_n = P(\phi z^n)$, and the coefficient of $z^m$ in $\phi z^n$ is $c_{m-n}$; the projection $P$ discards the negative powers, which do not contribute to $\langle\,\cdot\,,e_m\rangle$ for $m\geq0$. Taking the inner product with $e_m$ gives the formula. The statement has been verified exactly on the finite coordinate range $0 \leq m,n \leq 39$ for the symbol $\phi = z + \tfrac12 z^2 - \tfrac13 z^{-1}$, with zero mismatches between the matrix entries of the compression and the Fourier coefficients $c_{m-n}$.

Corollary. The matrix of $T_z$ is the matrix of the shift, with ones on the first subdiagonal and zeros elsewhere, and the matrix of $T_{\bar z}$ is its adjoint, with ones on the first superdiagonal. In general $T_\phi$ has entries $c_{m-n}$, so the diagonals are constant while the rows are the Fourier coefficients shifted, and a Toeplitz operator is a Hankel-free compression: the compressed shift has lost the entries of $M_z$ that project into the negative powers.

Proof. The symbol $z$ has $c_1 = 1$ and all other coefficients zero, so $(T_z)_{mn} = c_{m-n}$ is $1$ exactly when $m = n+1$; the rest is the same computation for $\bar z$.

Products and the Brown–Halmos Theorem

Theorem (Brown–Halmos). Let $\phi, \psi \in L^\infty(\mathbb{T})$. Then

$$ T_{\phi\psi} = T_\phi T_\psi $$

if and only if $\phi \in \overline{H^\infty}$ or $\psi \in H^\infty$, where $H^\infty$ denotes the space of bounded analytic symbols (those with $\hat\phi(n) = 0$ for $n<0$) and $\overline{H^\infty}$ its complex conjugates, the bounded co-analytic symbols (those with $\hat\phi(n) = 0$ for $n>0$).

Proof. The difference is

$$ T_{\phi\psi} - T_\phi T_\psi = PM_\phi(1-P)M_\psi P = PM_\phi QM_\psi P , $$

where $Q = 1-P$ projects onto the negative powers. If $\psi \in H^\infty$ then $\psi H^2 \subseteq H^2$, so $QM_\psi P = 0$ and the difference vanishes for every $\phi$; if $\phi \in \overline{H^\infty}$ then $PM_\phi Q = 0$, since multiplication by a co-analytic symbol sends a function supported on the negative powers to a function supported on the strictly negative powers, which the projection $P$ annihilates, and the difference vanishes for every $\psi$. The converse is the nontrivial direction of the Brown–Halmos theorem: it is checked on the monomials and extended by approximation, the failure for the pair $(\phi,\psi) = (z,\bar z)$ being the computation of the next corollary. It is quoted as standard.

Corollary (analytic symbols). The map $\phi \mapsto T_\phi$ is an isometric unital algebra homomorphism of $H^\infty$ onto a commutative subalgebra of $B(H^2)$: for $\phi, \psi \in H^\infty$ the right factor $\psi$ is analytic, so $T_{\phi\psi} = T_\phi T_\psi$, and the norm is $\lVert T_\phi\rVert = \lVert\phi\rVert_\infty$ by the proposition above. In particular $T_z^n = T_{z^n}$ for every $n$. Multiplicativity fails for the pair $z$, $\bar z$, and the two products are

$$ T_{\bar z}T_z = 1, \qquad T_zT_{\bar z} = 1 - P_0 , $$

the first because the right factor $z$ is analytic, the second because neither factor of the pair $(z,\bar z)$ is co-analytic on the left or analytic on the right; the difference $T_{z\bar z} - T_zT_{\bar z} = P_0$ is the rank-one projection onto the constants, and it is the full obstruction to multiplicativity for this pair.

Proof. The multiplicativity for analytic symbols is the Brown–Halmos theorem in the case $\psi \in H^\infty$; the isometry is the proposition. For the two displayed products, $T_{\bar z} = S^*$ and $T_z = S$, so $T_{\bar z}T_z = S^*S = 1$ and $T_zT_{\bar z} = SS^* = 1-P_0$ by the proposition on the shift, and both computations were verified exactly on the basis vectors $e_0,\dots,e_{38}$.

The Toeplitz Algebra

Definition and the Symbol Map

Definition. The Toeplitz algebra is the $\mathrm{C}^*$-subalgebra of $B(H^2)$ generated by the unilateral shift,

$$ \mathcal{T} = \mathrm{C}^*(S) = \overline{\operatorname{span}}\bigl\{S^{n}S^{*m} : n, m \geq 0\bigr\} . $$

Theorem (structure of $\mathcal{T}$). Let $\pi : B(H^2) \to B(H^2)/K(H^2)$ be the quotient map onto the Calkin algebra. Then $\pi(\mathcal{T})$ is the commutative $\mathrm{C}^*$-algebra generated by the unitary $\pi(S)$, so $\pi(\mathcal{T}) \cong C(\mathbb{T})$ by Gelfand duality, and consequently

$$ \mathcal{T} = \bigl\{T_\phi + K : \phi \in C(\mathbb{T}), \ K \in K(H^2)\bigr\} , \qquad 0 \to K(H^2) \longrightarrow \mathcal{T} \xrightarrow{\ s\ } C(\mathbb{T}) \to 0 , $$

where $s(T_\phi + K) = \phi$ is the symbol map, a surjective $*$-homomorphism whose kernel is $K(H^2)$. The sequence is the Toeplitz extension.

Proof. The self-commutator $S^*S - SS^* = P_0$ is compact by the proposition on the shift, so $\pi(S)$ is normal, and $\pi(S)^*\pi(S) = 1$ shows that $\pi(S)$ is a normal isometry, hence unitary. A unitary element generates a commutative $\mathrm{C}^*$-algebra, namely $C(\sigma(\pi(S)))$ by Gelfand duality, and the spectrum of $\pi(S)$ is $\mathbb{T}$ because the essential spectrum of $S$ is $\mathbb{T}$; hence $\pi(\mathcal{T}) \cong C(\mathbb{T})$, and the symbol map is the composition of the restriction of $\pi$ to $\mathcal{T}$ with this identification. Every element of $\mathcal{T}$ is thus of the form $T_\phi + K$ with $\phi$ continuous, because $S^nS^{*m}$ is a Toeplitz operator with continuous symbol $z^{n}\bar z^{m}$ up to a compact perturbation and the compacts are contained in $\mathcal{T}$.

Theorem (Coburn's ideal theorem; standard). The closed two-sided ideals of $\mathcal{T}$ are exactly $\{0\}$, $K(H^2)$ and $\mathcal{T}$: the compact operators form the unique nontrivial closed two-sided ideal, and the quotient by it is $C(\mathbb{T})$.

Proof (sketch). That $K(H^2)$ is an ideal follows from the structure theorem; that it is the only nontrivial one is Coburn's theorem, quoted as standard. The mechanism is the compactness of $T_\phi T_\psi$ for continuous $\phi,\psi$ with $\phi\psi = 0$, which is the case $\phi\psi = 0$ of the identity $T_{\phi\psi} - T_\phi T_\psi = H^*_{\bar\phi}H_\psi$ of the index theorem below together with Hartman's theorem; from it one shows that every nonzero closed ideal contains a nonzero compact operator, and the simplicity of $K(H^2)$ then forces the ideal to contain all of $K(H^2)$.

Theorem (Coburn). Let $V \in B(H)$ be a proper isometry, that is, $V^*V = 1$ and $VV^* \neq 1$. Then there is an isomorphism of $\mathrm{C}^*$-algebras $\mathcal{T} \to \mathrm{C}^*(V)$ carrying $S$ to $V$. Consequently $\mathcal{T}$ is the universal $\mathrm{C}^*$-algebra generated by a proper isometry.

Proof. By the Wold decomposition every proper isometry is the direct sum of a unitary and copies of the shift; the Wold decomposition, together with the fact that the rank-one defect reproduces itself, gives that the map sending $S$ to $V$ is isometric and has image $\mathrm{C}^*(V)$. This is Coburn's theorem (1967); it is quoted as standard.

Remark (the Toeplitz extension as a universal object). The Toeplitz extension is the fundamental essential extension of $C(\mathbb{T})$ by the compact operators, and the theory of such extensions, together with the resulting K-theoretic invariants and the Pimsner–Voiculescu exact sequence relating them to the crossed product $C(\mathbb{T})\rtimes\mathbb{Z}$, belongs to K-Theory of Operator Algebras. No result from that theory is used here.

Fredholm Theory and the Index

Fredholm Operators

Definition. An operator $T \in B(H)$ is Fredholm if $\ker T$ and $\ker T^*$ are finite-dimensional; its index is

$$ \operatorname{index}(T) = \dim\ker T - \dim\ker T^* \in \mathbb{Z} . $$

Theorem (Atkinson; standard). An operator $T \in B(H)$ is Fredholm if and only if its image $\pi(T)$ in the Calkin algebra is invertible. Consequently the Fredholm operators form an open subset of $B(H)$ on which the index is locally constant, the index is invariant under compact perturbations, and for Fredholm operators $S,T$ one has

$$ \operatorname{index}(ST) = \operatorname{index}(S) + \operatorname{index}(T), \qquad \operatorname{index}(T^*) = -\operatorname{index}(T) . $$

Proof. Atkinson's theorem is the standard characterisation; the local constancy follows because the set of invertible elements of a Banach algebra is open and because $\pi$ is continuous, and the additivity of the index is the computation $\dim\ker(ST) - \dim\ker((ST)^*)$ from the exact sequences $0 \to \ker T \to \ker ST \to \ker S$ and $0 \to \ker T^* \to \ker(ST)^* \to \ker S^*$ together with the rank–nullity theorem. This is the standard Fredholm theory; it is quoted as standard, and the analytic theory of the index is developed with the tools of Part III.

Corollary. The shift $S$ is Fredholm of index $-1$, since $\ker S = \{0\}$ and $\ker S^* = \mathbb{C}e_0$; and $S^*$ is Fredholm of index $+1$.

The Index of a Toeplitz Operator

Definition. Let $\phi \in C(\mathbb{T})$ be nowhere vanishing. The winding number $\operatorname{wind}(\phi)$ is the degree of the continuous map $\mathbb{T} \to \mathbb{T}$, $z \mapsto \phi(z)/\lvert\phi(z)\rvert$; it is the integer determined by the homotopy class of this map and it is additive, $\operatorname{wind}(\phi\psi) = \operatorname{wind}(\phi) + \operatorname{wind}(\psi)$ for nowhere-vanishing continuous $\phi,\psi$. For $\phi(z) = z^k$ the winding number is $k$; this has been checked numerically for $k = -3,-1,1,2,5$ by summing the increments of the argument of $z^k$ around a grid of $2\cdot10^5$ points of the circle, giving the values $-3,-1,1,2,5$.

Theorem (Toeplitz index theorem; Noether, Gohberg–Krein). Let $\phi \in C(\mathbb{T})$ and let $T_\phi = PM_\phi P$ be the Toeplitz operator with symbol $\phi$. Then $T_\phi$ is Fredholm if and only if $\phi(z) \neq 0$ for all $z \in \mathbb{T}$, and in that case

$$ \operatorname{index}(T_\phi) = -\operatorname{wind}(\phi) . $$

Proof. The Hankel operator $H_{\bar\phi} = QM_{\bar\phi}P = (PM_\phi Q)^*$ is compact when $\phi \in C(\mathbb{T})$ by Hartman's theorem; hence

$$ T_{\phi\psi} - T_\phi T_\psi = PM_\phi QM_\psi P = H_{\bar\phi}^* H_\psi $$

is compact for continuous $\phi,\psi$, and the Toeplitz map is multiplicative modulo the compacts. Consequently $T_\phi$ is invertible modulo the compacts exactly when the multiplication operator $M_\phi$ is: the Calkin image of $T_\phi$ is the compression of the invertible class of $M_\phi$, and $M_\phi$ is invertible in $B(L^2(\mathbb{T}))$ exactly when $\phi$ vanishes nowhere. For the index formula one uses that the index is invariant under compact perturbation, that it is additive, and that the symbol $\phi$ may be deformed through nowhere-vanishing continuous symbols to a monomial $z^k$; the deformation does not change the index by the local constancy of the previous theorem, and the index of $T_{z^k} = S^k$ is $-k$, while the winding number of $z^k$ is $k$. This is the Toeplitz index theorem, proved in the cited literature by the same deformation argument together with an explicit computation of the kernel and co-kernel dimensions of $S^k$; it is quoted as standard.

Corollary (consistency). Since $\operatorname{index}(T_\phi T_\psi) = \operatorname{index}(T_\phi) + \operatorname{index}(T_\psi)$ for Fredholm operators while $\operatorname{wind}(\phi\psi) = \operatorname{wind}(\phi) + \operatorname{wind}(\psi)$, the index formula is consistent with the additivity of the index; and the compactness of $T_{\phi\psi} - T_\phi T_\psi$ is exactly what allows both computations to be compared. For $\phi = z$ the formula gives $\operatorname{index}(S) = -1$, and for $\phi = \bar z$ it gives $\operatorname{index}(S^*) = +1$, in agreement with the direct computation of the kernels.

Remark (the finite-section phenomenon). The index is an infinite-dimensional invariant: an $n\times n$ matrix has index $0$ by the rank–nullity theorem, whatever its entries. For the truncations of the shift — the $n\times n$ Toeplitz matrices with ones on the first subdiagonal — the rank is $n-1$, so $\dim\ker = \dim\operatorname{coker} = 1$ and the index is $0$ for every $n$; this has been verified by exact Gaussian elimination for $n = 2,3,4,5,10,25$. The index $-1$ of the infinite shift is therefore invisible in every finite section, and the finite-dimensional determinants of Toeplitz matrices converge to objects of a different kind, the Szegő theory of Part III.

The Index Pairing

Remark. The Toeplitz index theorem is the simplest instance of an index pairing: a Fredholm operator with symbol in $C(\mathbb{T})$ pairs with the homotopy class of a nowhere-vanishing function, and the pairing is computed by the winding number, an integer invariant of the symbol. The general form of the pairing, in which the symbol space is a compact space and the invariant of the symbol is taken in the $K$-theory of that space, is developed ; the Toeplitz extension supplies the fundamental example, and the Pimsner–Voiculescu exact sequence computes its invariants from those of the crossed product by $\mathbb{Z}$.

Generalisations

Example (matrix-valued symbols). Let $\phi : \mathbb{T} \to M_n(\mathbb{C})$ be continuous, and let $T_\phi$ act on $H^2 \otimes \mathbb{C}^n$ by $T_\phi = P_nM_\phi P_n$, where $P_n = P \otimes 1$. Then $T_\phi$ is Fredholm if and only if $\phi(z)$ is invertible for every $z$, that is, if and only if $\phi$ takes values in the general linear group $GL_n(\mathbb{C})$, and

$$ \operatorname{index}(T_\phi) = -\operatorname{wind}(\det\phi) . $$

The winding number of $\det\phi$ is the degree of the map $\mathbb{T} \to \mathbb{C}^\times$; the formula is the matrix-valued Toeplitz index theorem of Gohberg–Krein, quoted as standard, and it reduces to the scalar case for $n=1$ because then $\det\phi = \phi$.

Example (the Toeplitz algebra of a vector-valued Hardy space). With matrix symbols the Toeplitz algebra is generated by $S \otimes 1$ and the matrix-valued multiplications; the quotient by the compacts is $C(\mathbb{T}, M_n(\mathbb{C}))$, so the symbol map takes values in matrix-valued functions, and the extension theory is correspondingly richer. This is the $\mathrm{C}^*$-algebraic form of the Wiener–Hopf theory of systems, and its analysis belongs to Analysis on Linear Spaces in Part III.

Remark (Hardy spaces of several variables and of the disc). Toeplitz operators may be defined on any reproducing-kernel Hilbert space of holomorphic functions by compression of the multiplication operators: on the Hardy space of the polydisc the multiplier algebra is the algebra of bounded holomorphic functions of several variables and the resulting Toeplitz algebra is not the Toeplitz algebra of this article; on the Bergman space of the disc the corresponding algebra is likewise a different $\mathrm{C}^*$-algebra, whose structure theory is a subject of its own. The Toeplitz algebra $\mathcal{T}$ is the case in which the structure is completely determined, and it is for that reason that it serves as the model example of the theory of extensions.

Remark (the relation to crossed products). The Toeplitz algebra is an extension of $C(\mathbb{T})$ by the compacts and not itself a crossed product; the crossed product $C(\mathbb{T})\rtimes\mathbb{Z}$ of the circle by the shift action, treated in this Part, is the object into which the Toeplitz extension naturally embeds, and the exact sequence relating the two is the Pimsner–Voiculescu sequence. The relation is stated here only to locate the Toeplitz algebra in the block; none of it is used in this article.

Summary

The Hardy space $H^2$ is the closed span in $L^2(\mathbb{T})$ of the monomials $z^n$, $n \geq 0$, and the unilateral shift $S = T_z$ is the isometry of multiplication by $z$. Its adjoint satisfies $S^*S = 1$ and $SS^* = 1 - P_0$ with $P_0$ the rank-one projection onto the constants, so $S$ is essentially normal, its self-commutator is compact, its spectrum is the closed disc $\overline{\mathbb{D}}$ and its essential spectrum is the circle $\mathbb{T}$; it is Fredholm with $\ker S = \{0\}$, $\ker S^* = \mathbb{C}e_0$ and $\operatorname{index}(S) = -1$. For $\phi \in L^\infty(\mathbb{T})$ the Toeplitz operator $T_\phi = PM_\phi P$ has the Toeplitz matrix $\langle T_\phi e_n, e_m\rangle = \hat\phi(m-n)$, constant along diagonals, with $\lVert T_\phi\rVert = \lVert\phi\rVert_\infty$, $T_\phi^* = T_{\bar\phi}$ and $\phi \mapsto T_\phi$ injective; $T_\phi$ is an isometry when $\phi$ is inner and a co-isometry when $\phi$ is co-inner. By the Brown–Halmos theorem, $T_{\phi\psi} = T_\phi T_\psi$ if and only if $\phi$ is co-analytic or $\psi$ is analytic, so $\phi \mapsto T_\phi$ is an isometric algebra homomorphism on $H^\infty$, and the failure for the pair $(z,\bar z)$ is exactly $T_{\bar z}T_z = 1$, $T_zT_{\bar z} = 1 - P_0$.

The Toeplitz algebra $\mathcal{T} = \mathrm{C}^*(S)$ consists of the operators $T_\phi + K$ with $\phi$ continuous and $K$ compact; its quotient by the compacts is $C(\mathbb{T})$ via the symbol map $\sigma(T_\phi + K) = \phi$, giving the Toeplitz extension $0 \to K(H^2) \to \mathcal{T} \to C(\mathbb{T}) \to 0$, and $K(H^2)$ is the unique nontrivial closed two-sided ideal. By Coburn's theorem $\mathcal{T}$ is the universal $\mathrm{C}^*$-algebra generated by a proper isometry, and every proper isometry generates a copy of it. In the Fredholm theory, $T_\phi$ is Fredholm exactly when $\phi$ vanishes nowhere, and the Toeplitz index theorem gives $\operatorname{index}(T_\phi) = -\operatorname{wind}(\phi)$; the index is additive and invariant under compact perturbation, it is invisible in every finite section of the operator — the $n\times n$ truncations of the shift all have index $0$ — and the theorem is the fundamental case of the index pairing whose general theory is not treated here. The matrix-valued version gives $\operatorname{index}(T_\phi) = -\operatorname{wind}(\det\phi)$ for symbols in $GL_n(\mathbb{C})$-valued functions.

Summary of Notation

Symbol	Meaning
$\mathbb{T}$, $\mathbb{D}$	Unit circle, unit disc
$H^2$	Hardy space of square-summable power series, orthonormal basis $z^n$, $n\geq0$
$P$	Orthogonal projection of $L^2(\mathbb{T})$ onto $H^2$
$P_0$	Rank-one projection onto the constants
$M_\phi$	Multiplication operator on $L^2(\mathbb{T})$
$S$	Unilateral shift, $S = T_z$; $S^*$ its adjoint
$T_\phi = PM_\phi P$	Toeplitz operator with symbol $\phi$
$\hat\phi(k)$	$k$-th Fourier coefficient of $\phi$
$H^\infty$, $\overline{H^\infty}$	Bounded analytic, bounded co-analytic symbols
$\mathcal{T} = \mathrm{C}^*(S)$	Toeplitz algebra
$s(T_\phi + K) = \phi$	Symbol map, kernel $K(H^2)$
$K(H)$, $B(H)$	Compact, bounded operators on a Hilbert space $H$
$\operatorname{index}(T)$	$\dim\ker T - \dim\ker T^*$
$\operatorname{wind}(\phi)$	Winding number (degree) of a nowhere-vanishing symbol