Toeplitz extensions in noncommutative topology and mathematical physics

We review the theory of Toeplitz extensions and their role in operator K-theory, including Kasparov's bivariant K-theory. We then discuss the recent applications of Toeplitz algebras in the study of solid state systems, focusing in particular on the bulk-edge correspondence for topological insulators.

The paper is structured as follows. In Section 2 we review the construction of the classical one-dimensional Toeplitz algebra as the universal C * -algebra generated by a single isometry, and we recall its role in the Noether-Gohberg-Krein index theorem, which relates the index of Toeplitz operators to the winding number of their symbol. We conclude the section by discussing how the construction can be extended to higher dimensions. In Section 3 we take a deep dive into the world of noncommutative topology and discuss the role of Toeplitz extensions in operator K-theory, namely in Cuntz's proof of Bott periodicity and in the development of Kasparov's bivariant K-theory. This rather technical section allows us to introduce the tools that are needed in the noncommutative approach to solid state physics. In Section 4, we describe two constructions of universal C * -algebras that will later play a crucial role in the study of solid state systems, namely crossed products by the integers, Cuntz-Pimsner algebras, and their Toeplitz algebras. Finally, Section 5 is devoted to describing how Toeplitz extensions and the associated maps in K-theory provide the natural framework for implementing the bulk-edge correspondence from solid state physics.

Toeplitz algebras of operators
2.1. Shifts, winding numbers, and the Noether-Gohberg-Krein index theorem. In view of the Gelfand-Naimark theorem [21], every abstract C * -algebra, commutative or not, admits a faithful representation as a subalgebra of the algebra B(H) of bounded operators on some Hilbert space H. In this section, we will start by constructing two concrete examples of C * -algebras of operators. As mentioned in the Introduction, we are interested in how the commutative algebra of functions on the circle and the noncommutative algebra generated by a single isometry fit together in a short exact sequence. This extension will later serve as our prototypical example illustrating the use of C * -algebraic techniques in solid state physics.
Let S 1 := {z ∈ C | zz = 1} denote the unit circle in the complex plane. The corresponding C * -algebra, C(S 1 ), is the closure in the supremum norm of the algebra of Laurent polynomials O(S 1 ) = C[z, z] zz = 1 .
The algebra C(S 1 ) admits a convenient representation on the Hilbert space L 2 (S 1 ) of square-integrable functions on S 1 . This Hilbert space is isomorphic to the Hilbert space of sequences ℓ 2 (Z), and the isomorphism is implemented by the discrete Fourier transform Under this isomorphism, the operator of multiplication by z is mapped to the bilateral shift operator U , defined on the standard basis {e n } n∈Z of ℓ 2 (Z) via (2) U (e n ) = (e n+1 ), U * (e n ) = e n−1 .
It is easy to see that U is a unitary operator, i.e. U * U = 1 = U U * . The algebra C(S 1 ) is then isomorphic to the smallest C * -subalgebra of B(ℓ 2 (Z)) that contains U . In order to define the second C * -algebra we are interested in, which is genuinely non-commutative, we shall consider the Hardy space H 2 (S 1 ). This is defined as the subset of L 2 (S 1 ) consisting of holomorphic L 2 -functions. The discrete Fourier transform allows us to identify the Hardy space with the sequence space ℓ 2 (N). We will denote by p the orthogonal projection from ℓ 2 (Z) to ℓ 2 (N), and by P that from L 2 (S 1 ) onto H 2 (S 1 ) (obtained by conjugating p with the Fourier transform).
Multiplication by z on the Hardy space corresponds to a shift operator on ℓ 2 (N), called the unilateral shift, expressed on the standard basis {f n } n∈N of ℓ 2 (N) via: Its adjoint is not invertible, as This motivates the following: It is easy to see that the Toeplitz algebra T is not commutative, as In particular, it follows from (3) that elements of T commute up to compact operators, and in particular the generator T is unitary module compact operators. In other words, the Toeplitz algebra can be viewed as the C * -algebra extension of continuous functions on the circle by the compact operators: The extension (4) admits a completely positive and completely contractive splitting given by the Hardy projection. Indeed, for every f ∈ C(S 1 ), the assignment defines a bounded operator on the Hardy space H 2 (S 1 ), where, under Fourier transform, T z corresponds to the unilateral shift. As the function z generates C(S 1 ) as a C * -algebra, every such T f is an element of T .
The following result implies that the Toeplitz algebra is the universal C * -algebra generated by an element T satisfying T * T = 1: [12]). Suppose v is an isometry in a unital C * -algebra A. Let T = T z ∈ T . Then there exists a unique unital * -homomorphism φ : T → A such that φ(T ) = v. Moreover, if vv * = 1, then the map φ is isometric.
2.1.1. The Noether-Gohberg-Krein index theorem. Recall that an operator F ∈ B(H) is a Fredholm operator if both ker F and ker F * are finite-dimensional. The Fredholm index of such an operator is the integer One of the key properties of the Fredholm index is that it is constant along continuous paths of Fredholm operators. As such it is a homotopy invariant.
The completely positive linear splitting f → T f allows one to give a precise characterization of which Toeplitz operators T f are Fredholm. Moreover, the index of a Fredholm Toeplitz operator T f can be described entirely in terms of a familiar homotopy invariant of the complex function f . This is the content of the Toeplitz index theorem, due to F. Noether and later reproved independently by Gohberg and Krein. It was one of the first results linking index theory to topology and should be viewed as an ancestor to the celebrated Atiyah-Singer index theorem. Theorem 2.3 (Noether [33], Gohberg-Krein [23]). For f : S 1 → C × the operator T f : H 2 (S 1 ) → H 2 (S 1 ) is Fredholm and with w(f ) the winding number of f . If f is a C 1 -function, then the winding number can be computed as The latter, explicit expression for the winding number and hence the Toeplitz index should be viewed as a result of differential topology: By choosing a nice representative in the homotopy class of the function f , the differential calculus can be employed to compute a topological invariant. We will see an application of this computation in Section 5.
For every z ∈ ∂Ω, the Levi form , z is defined as Then Ω is called a strongly pseudo-convex domain if the Levi form is positive semi-definite on the complex tangent space at every point z ∈ ∂Ω, i.e., u ∈ T z (∂Ω), u = 0 implies u, u z > 0, Open balls in C n are examples of strongly pseudo-convex domains. However, the product of two open balls is not strongly pseudo-convex, showing the notion is somewhat subtle.
Given a strongly pseudo-convex domain Ω ⊆ C n with smooth boundary, we denote by L 2 (∂Ω) the Hilbert space of square integrable functions on the boundary ∂Ω. The Hardy space H 2 (∂Ω) is defined as the Hilbert space closure in L 2 (∂Ω) of boundary values of homolomorphic functions on Ω that admit a continuous extensions to the boundary ∂Ω (cf. [40,Definition 2.3]). The orthogonal projection called the Cauchy-Szegö projection, is used to define Toeplitz operators, in analogy with (5). Indeed, let f be a continuous function on ∂Ω, the Toeplitz operator with symbol f is defined as for all g ∈ H 2 (∂Ω).
For any two f, f ′ ∈ C(∂Ω), the product of Toeplitz operators T f • T f ′ is equal to T f f ′ modulo compact operators. Moreover, for any f ∈ C(∂Ω), the operator T f is compact if and only if f is identically zero. These two facts combined lead to the following: Let Ω be a strictly pseudo-convex domain. There is an extension of C * -algebras: The extension admits a completely positive and completely contractive linear splitting given by the Cauchy-Szegö projection.
Applied to the unit ball in C n this construction yields the Toeplitz extensions for odd-dimensional spheres as a special case: which clearly recover (4) for d = 1.
The Toeplitz algebra T (S 2d−1 ) admits an equivalent description in terms of socalled d-shifts, as described in [2,Theorem 5.7]. For an overview of the interplay of Toeplitz C * -algebras and index theory, as well as their role in the computation of noncommutative invariants, we refer the reader to the excellent survey [32].

Toeplitz algebras in operator K-theory and bivariant K-theory
Operator K-theory is a functor, associating to a C * -algebra A two Abelian groups K * (A), * = 0, 1. Functoriality means that for a * -homomorphism ϕ : A → B between C * -algebras A and B, there is an induced homomorphism of Abelian groups The key properties of the operator K-theory functor are that it is homotopy invariant, half-exact and Morita invariant. We now define each of these properties more precisely.
Homotopy invariance is the property that if ϕ and ψ are connected by a continuous path of * -homomorphisms, then the induced maps on K-theory coincide, that is ϕ * = ψ * .
Half-exactness is the property that for any extension of C * -algebras the corresponding sequence of groups Lastly, Morita invariance entails that for any rank-one projection p ∈ K = K(ℓ 2 (N)), the * -homomorphism induces an isomorphism in K-theory.
Recall that the suspension SA of a C * -algebra A is defined to be which is a C * -algebra in the sup-norm, and pointwise product and involution inherited from A. The operation A → SA is functorial for * -homorphisms, and it is customary to define the higher K-groups as K n (A) := K 0 (S n A). Via a general construction in topology, it follows that the extension (6) induces a long exact sequence of Abelian groups.
The boundary maps in such exact sequences are often related to index theory. For instance, for the Toeplitz extension (4), the boundary map maps the class of a nonzero function f ∈ C(S 1 ) to the index of the Toeplitz operator T f . One of the key features of operator K-theory is Bott periodicity. It states that for any C * -algebra A there are natural isomorphisms between its K-theory and the K-theory of its double suspension S 2 A. It turns out that the three properties of homotopy invariance, half-exactness and Morita invariance suffice to deduce the existence of natural Bott periodicity isomorphisms K * (A) ≃ K * (S 2 A). As a consequence, there are only two K-functors, K 0 and K 1 , and the exact sequence (7) reduces the cyclic six-term exact sequence Cuntz's proof of Bott periodicity. Apart from the invariance properties of the K-functor, Cuntz's proof of Bott periodicity (cf. [18]) exploits essential properties of the Toeplitz extension (4). By composing the projection homomorphism π : T → C(S 1 ) with the evaluation map ev 1 : we obtain a character of T : The unital embedding ι : C → T splits the homomorphism χ in the sense that χ • ι = id C . It is a non-trivial fact these * -homomorphisms are mutually inverse in K-theory, in a strong sense made precise below.
To state the result, which lies at the heart of the proof of the Bott periodicity theorem, we shall recall the construction of the spatial or minimal tensor product A 1 ⊗A 2 of C * -algebras A i , i = 1, 2. Choose faithful representations π i : A i → B(H i ) and let H 1 ⊗ H 2 be the completed tensor product of Hilbert spaces. One defines A⊗B to be the completion of the algebraic tensor product A ⊗ B in the norm inherited from the representation Tensor products of C * -algebras are not unique, and the spatial tensor product is the completion in the minimal C * -norm on the algebraic tensor product A ⊗ B. There is also a maximal C * -norm on A ⊗ B, which involves taking the supremum over all representations. A C * -algebra N is nuclear, if for any other C * -algebra A, the minimal and maximal C * -tensor norms on N ⊗ A coincide. For our purposes it suffices to know that all commutative C * -algebras are nuclear. Given an extension of C * -algebras the sequence of tensor products may fail to be exact in the middle. However, nuclearity of the C * -algebra B guarantees exactness.
We can now exploit Proposition 3.1, Lemma 3.2, and the exactness properties of the K-functor to deduce Bott periodicity.
Proof. Consider the character χ defined in (9) and let T 0 := ker χ, so that we have an extension 0 As C is nuclear, this extension has the property that the induced sequence is exact for any C * -algebra A as well, by Lemma 3.2.
The long exact sequence (7), together with the fact that S(A⊗B) ≃ A⊗SB and Proposition 3.1, imply that χ * : K n (T ⊗A) → K n (A) is an isomorphism for all n. Consequently K n (T 0 ⊗A) = 0 for all n. Now observe that, after identifying ker ev 1 with C 0 (0, 1), we can construct a second extension As C 0 (0, 1) is nuclear, this extension, too, has the property that is exact for any C * -algebra A, by Lemma 3.2. Since C 0 (0, 1)⊗A ≃ SA, the long exact sequence (7) gives an isomorphism Now we use the Morita invariance isomorphism K n (K⊗A) ≃ K n (A) and the fact that C(0, 1)⊗A ≃ SA to deduce that which yields the Bott periodicity isomorphism.
We remark that, in fact, the theorem holds if we replace K by any functor that is homotopy invariant, half-exact and Morita invariant.
3.2. Toeplitz extensions and bivariant K-theory. As we have seen so far in the Toeplitz index and Bott periodicity theorems, extensions of C * -algebras play a crucial role in K-theory and henceforth in index theory. An extension of a C*algebra A by B should be viewed as a new C * -algebra, built by "gluing together" A and B in a possibly topologically nontrivial way.
In [10], Brown, Douglas, and Fillmore initiated the study of extensions by considering exact sequences of the form for some Hilbert space H and some compact Hausdorff topological space M . They proved that such extensions form an Abelian group by defining addition via an appropriate version of the Baer sum. They also showed that their Abelian group is dual to K-theory in a precise sense governed by Fredholm index theory. Kasparov generalized this construction to extensions where A is a separable C * -algebra and X a countably generated Hilbert C * -module over a second, σ-unital C * -algebra B. A technical assumption on such extensions is that they admit a completely positive and completely contractive linear splitting ℓ : A → E such that ℓ•π = id A . This assumption is automatically satisfied when the quotient algebra in the extension is nuclear. Commutative C * -algebras are nuclear, and thus the Toeplitz extensions discussed previously satisfy this assumption. The isomorphism classes of such extensions form an Abelian group Ext 1 (A, B) which is isomorphic to the Kasparov group KK 1 (A, B). This section is devoted to making this statement more precise. An excellent reference for this discussion is [24, Chapter 3].
3.2.1. Hilbert modules and C * -correspondences. Before we proceed, we need to recall some results from the theory of Hilbert C * -modules. For more details on the latter, we refer the interested reader to the monograph [31].

Definition 3.4.
A pre-Hilbert module over B is a right B-module X with a Bvalued Hermitian product, i.e. a map ·, · B : X × X → B satisfying for all ξ, η ∈ X and for all b ∈ B.
For a pre-Hilbert module X, one can define a scalar valued norm · using the C * -norm on B: Definition 3.5. A Hilbert C * -module is a pre-Hilbert module that is complete in the norm (12).
If one defines X, X to be the linear span of elements of the form ξ, η for ξ, η ∈ X, then its closure its a two-sided ideal in B. We say that the Hilbert module X is full whenever X, X is dense in B.
Let now X, Y be two Hilbert C * -modules over the same C * -algebra B.
Definition 3.6. A map T : X → Y is said to be an adjointable operator if there exists another map T * : Y → X with the property that T ξ, η = ξ, T * η for all ξ ∈ X, η ∈ Y .
Every adjointable operator is automatically right B-linear and bounded. However, the converse is in general not true: a bounded linear map between Hilbert modules need not be adjointable.
We denote the collection of adjointable operators from X to Y by End * B (X, Y ). When X = Y , the adjointable operators form a C * -algebra in the operator norm, that is denoted by End * B (X). Inside the adjointable operators one can single out a particular subspace, which is analogous to that of finite-rank operators on a Hilbert space. More precisely, for every ξ ∈ Y, η ∈ X one defines the operator θ ξ,η : X → Y as (13) θ ξ,η (ζ) = ξ η, ζ , ∀ζ ∈ X This is an adjointable operator, with adjoint θ * ξ,η : Y → X given by θ η,ξ . We denote by K B (X, Y ) the closure of the linear span of (14) {θ ξ,η | ξ, η ∈ X} ⊆ End * B (X, Y ), and we refer to it as the space of compact adjointable operators.
In particular K B (X) := K B (X, X) ⊆ End * B (X) is a closed two-sided ideal in the C * -algebra End * B (X), hence a C * -subalgebra, whose elements are referred to as compact endomorphisms. Elements of K B (X) and of End * B (X) act on X from the left, motivating the following: we refer to (X, φ) as a compact C * -correspondence and in case A = B we refer to (X, φ) as a C * -correspondence over B.
When no confusion arises, we will omit the map φ and simply write X.
Two C * -correspondences X φ and Y ψ over the same algebra B are called isomorphic if and only if there exists a unitary U ∈ End * B (X, Y ) intertwining φ and ψ.
Given an (A, B)-correspondence X φ and a (B, C)-correspondence Y ψ , one can construct an (A, C)-correspondence, named the interior tensor product of X φ and Y ψ .
As a first step, one constructs the balanced tensor product X ⊗ B Y which is a quotient of the algebraic tensor product X ⊗ alg Y by the subspace generated by elements of the form (15) ξb This has a natural structure of right module over C given by and a C-valued inner product defined on simple tensors as and extended by linearity. The inner product is well-defined (cf. [31, Proposition 4.5]); in particular, the null space N = {ζ ∈ X ⊗ alg Y ; ζ, η = 0} can be shown to coincide with the subspace generated by elements of the form in (15).
One then defines X ⊗ ψ Y to be the right Hilbert module obtained by completing X ⊗ B Y in the norm induced by (16).
Moreover for every T ∈ End * B (X), the operator defined on simple tensors by ξ ⊗ η → T (ξ) ⊗ η extends to a well-defined operator φ * (T ) := T ⊗ 1. It is adjointable with adjoint given by T * ⊗ 1 = φ * (T * ). In particular, this means that there is a left action of A defined on simple tensors by and extended by linearity to a map thus turning X ⊗ ψ Y into an (A, C)-correspondence. For all the details, we refer the reader once more to [31,Chapter 4]. We remark that the interior tensor product induces an associative operation on isomorphism classes of C * -correspondences.

3.2.2.
Kasparov modules and the theory of extensions. We now come to defining the key objects in Kasparov's bivariant K-theory [29], which are inspired by the geometry of elliptic operators on manifolds.
Definition 3.8. An odd Kasparov (A, B)-bimodule is a pair (Y, F ) where Y = Y φ is a Hilbert C * -correspondence from A to B, and F ∈ End * B (Y ) is a self-adjoint operator such that F 2 = 1 and [F, a] ∈ K(Y ). An even Kasparov module is a triple (Y, F, γ) such that (Y, F ) is an odd Kasparov module and γ ∈ End * B (Y ) is a self-adjoint unitary that commutes with A and anticommutes with F .
The natural equivalence relation of homotopy of Kasparov modules is conveniently defined via Kasparov modules for (A, C([0, 1], B)). The homotopy classes of odd Kasparov (A, B)-modules form an Abelian group denoted KK 1 (A, B). Similarly, the homotopy classes of even Kasparov modules form an Abelian group KK 0 (A, B). If we choose A = C then there are natural isomorphisms KK * (C, B) ≃ K * (B), and as such KK-theory generalises K-theory. The main feature of the theory is the existence of an associative, bilinear product structure the Kasparov product. Again, if we set A = C, we see that elements in KK j (B, C) induce maps K * (B) → K * +j (C) by taking products from the right. There is a close relationship between the Abelian groups KK 1 (A, B) and Ext 1 (A, B) which can be understood via the following Kasparov-Stinespring theorem, first proved in [28]. Theorem 3.9 (see the proof of Theorem 3.2.7 in [24]). Let A, B be C * -algebras, with A separable and B σ-unital. Let X be a countably generated Hilbert C * -module over B and ρ : A → End * B (X) be a completely positive contraction. There exists a countably generated Hilbert C * -module Y over B, a * -homomorphism π : A → End * B (Y ) and an isometry v : It is worth noting that such an isometry v : X → Y immediately gives rise to a Toeplitz type algebra To an extension with a completely positive linear splitting ℓ : A → E we can associate an odd Kasparov module by observing that, as K(X) is an ideal in E, there is a * -homomorphism ϕ : E → End * B (X). We consider the completely positive contraction ρ := ϕ • ℓ : A → End * B (X) and obtain an (A, B)-bimodule Y and an isometry v : X → Y via Theorem 3.9.
Theorem 3.10. Let X be a countably generated Hilbert C * -module over the σunital C * -algebra B and A a separable C * -algebra. If since for x 1 , x 2 ∈ X it holds that vθ x1,x2 v * = θ v(x1),v(x2) , and we compute This proves that (Y, F ) is a Kasparov module.
By the previous theorem, we see that an extension of C * -algebras induces an element in KK 1 (A, B). Using the product structure (17), this leads to the elegant viewpoint that an extension induces maps via the Kasparov product. These maps coincide with the boundary maps in the long exact sequence associated to the extension. For instance, the product with the extension of the previous section induces the Bott periodicity isomorphisms K n (S 2 A) ≃ K n (A). In fact, the extension above, in combination with the Kasparov product, can be used to prove the general bivariant Bott periodicity isomorphisms for any pair of separable C * -algebras (A, B). The Kasparov-Stinespring construction can be inverted up to homotopy, yielding the statement that KK 1 (A, B) is isomorphic to Ext 1 (A, B). Effectively, this amounts to the observation that KK-theory is nothing but the study of extensions of C * -algebras.
To conclude, let us sketch the inverse construction. An odd Kasparov module (X, F ) for (A, B) defines an adjointable projection P := 1 2 (F + 1) and hence a complemented submodule X := P Y ⊂ Y . The C * -subalgebra To see that E is closed under products, we use that P SP T P − P abP = P (S − a)P T P + P aP (T − b)P − P a(1 − P )bP which is an element of K(X). It admits the completely contractive linear splitting ℓ : A → E, ℓ : a → (P aP, a), and the inclusion K(X) = K(P Y ) → E defined by T → (T, 0) and the quotient map (P T P, a) → a with kernel K(X). The C * -algebra E can be viewed as an abstract Toeplitz algebra associated to the Kasparov module (Y, F ). This inverts the Kasparov-Stinespring construction, as is easily checked.

Toeplitz algebras, crossed products by the integers, and
Cuntz-Pimsner algebras We will now describe two constructions of Toeplitz C * -algebras and quotients thereof that appear in the study of solid state systems, as they provide the natural framework for implementing the bulk-edge correspondence.

4.1.
Crossed products by the integers and the Pimsner-Voiculescu Toeplitz algebra. Our first object of study are crossed products by the integers. They constitute one of the simplest and most well-understood examples of C * -dynamical systems, a class of objects which were introduced to study group actions on C *algebras.
Let B be a unital C * -algebra, and α ∈ Aut(B) a single automorphism. This defines an action of the additive group Z of integers on B given by The crossed product C * -algebra B ⋊ α Z is realised as the universal C * -algebra generated by B and a unitary u satisfying the covariance condition α n (b) = u n bu * n , ∀b ∈ B, n ∈ Z.
As described in [34], crossed products by a single automorphism can be realised as quotients in a Toeplitz exact sequence of C * -algebras, constructed starting from the Toeplitz extension (4). The Pimsner-Voiculescu Toeplitz algebra T (B, α) and the crossed product C *algebra B ⋊ α Z fit into a short exact sequence involving the stabilisation of B: Proof of exactness of the above sequence follows after tensoring the Toeplitz exact sequence (4) with the algebra B, using nuclearity of C(S 1 ) together with Lemma 3.2, and by realising B ⋊ α Z as a subalgebra of B⊗C(S 1 ) (see [34,Section 2]). The Pimsner-Voiculescu Toeplitz algebra T (B, α) is KK-equivalent to the algebra B itself. The exact sequence (18) then induces six-term exact sequences that allow for an elegant computation of the K-theory and K-homology groups of the crossed product algebra B ⋊ α Z in terms of those of the algebra B. These exact sequences are a special case of those described in Subsection 4.2.2.

4.2.
Pimsner's construction: universal C * -algebras from C * -correspondences. The construction which we shall describe now generalises that of crossed products by the integers. In [35], starting from a C * -correspondence (X, φ), Pimsner constructed two C * -algebras T X and O X , which are now referred to as the Toeplitz algebra and the Cuntz-Pimsner algebra of the pair (X, φ), respectively. Both algebras are characterized by universal properties and depend only on the isomorphism class of the pair (X, φ). We will describe the construction for compact correspondences. 4.2.1. The Toeplitz algebra. As one can take balanced tensor products of C * -corre spondences, as described in 3.2.1, we consider the modules and we take the infinite direct sum which is referred to as the (positive) Fock correspondence associated to the correspondence (X, φ).
One can naturally associate to any element ξ ∈ X a shift map: This is an adjointable operator on F X , with adjoint The Toeplitz algebra of the C * -correspondence X φ is the smallest C * -subalgebra of End * B (F X ) that contains all the T ξ for ξ ∈ X. When (X, φ) is a compact C * -correspondence, the compact operators on the Fock module sit inside T E as a two-sided ideal, motivating the following: Definition 4.3. The Cuntz-Pimsner algebra O X of a compact C * -correspondence (X, φ) is the quotient algebra appearing in the exact sequence The image of an element T ξ ∈ T X under the quotient map π will be denoted by S ξ .
Changing the ideal in the exact sequence (23), one can define the Cuntz-Pimsner algebra of a general (i.e. non-compact, and possibly non-injective) C * -correspondence. We will not be concerned with this more elaborate construction here. For details see [30,35] Many well-known and studied examples of C * -algebras admit a description as Toeplitz-Pimsner and Cuntz-Pimsner algebras. The theory provides a unifying framework for a variety of examples, ranging from the study of discrete dynamics to more geometric situations.
Example. Let B = C and X = C n and φ the left action by multiplication. If one chooses a basis for C n , then the Toeplitz algebra of (X, φ) is the universal C * -algebras generated by n isometries V 1 , . . . , V n satisfying i V i V * i ≤ 1. This yields the well known Toeplitz extension for the Cuntz algebras O n : where F is the full Fock space on C n . In particular, for n = 1 one gets back the classical Toeplitz extension of (4).
Example (cf. [25, Section 2])). If the correspondence X is a finitely generated and projective module over a unital C * -algebras, the Pimsner algebra of (X, φ) can be realized explicitly in terms of generators and relations. Indeed, since X is finitely generated and projective, there exists a finite set {η j } n j=1 of elements of X such that ξ = n j=1 η j η j , ξ B , ∀ξ ∈ X.
Then, using the above formula, one can spell out the left B-action on X as The C * -algebra O X is then the universal C * -algebra generated by B together with n operators S 1 , . . . , S n , satisfying Example. Let B be a C * -algebra and α : B → B an automorphism of B. Then X = B, seen as a module over itself, can be naturally made into a compact C *correspondence.
The right Hilbert B-module structure is the standard one, with right B-valued inner product a, b B = a * b. The automorphism α is used to define the left action via a · b = α(a)b and left B-valued inner product given by a, b Each module X (k) is isomorphic to B as a right-module, with left action (25) a · ( The corresponding Pimsner algebra O X coincides then with the crossed product algebra B ⋊ α Z, while the Toeplitz algebra T X agrees with the Toeplitz algebra T (B, α). The extension (23) then reduces to (18).

4.2.2.
Six-term exact sequences. As the Toeplitz extension (23) is semi-split whenever the coefficient algebra B is nuclear, it induces six-term exact sequences in KK-theory. These exact sequences can be simplified to a great extent after making the following observations: • For a compact C * -correspondence (X, φ), the triple(X, φ, 0) gives a welldefined even Kasparov module (with trivial grading), whose class we denote by [X]. • The ideal K(F X ) is naturally Morita equivalent to the algebra B itself.
• By [35,Theorem 4.4.], the Toeplitz algebra T X is KK-equivalent to the coefficient algebra B. In K-theory, the induced six-term exact sequence reads (26) K 0 (B) where i * is the map induced by the inclusion B ֒→ O X and the maps ∂ are connecting homomorphisms. Up to Morita equivalence, the latter can be computed as Kasparov products with the class of the extension (23). An unbounded representative for the extension class was constructed [22] in the setting bi-Hilbertian bimodules of finite Jones-Watatani index (cf. [26]), subject to some additional assumption. We conclude this section by remarking that, in the case of a self-Morita equivalence bimodule-i.e., whenever X is full and φ implements an isomorphism between B and K B (X)-the exact sequence (26) can be interpreted as a generalization of the classical Gysin sequence in K-theory (see [27,IV.1.13]) for the module of sections E of a noncommutative line bundle. The Kasparov product with the map 1 − [X] can be interpreted as a noncommutative Euler class. This analogy was exploited in [1] to compute K-theory groups of algebras presenting a circle bundle structure.

Applications to topological insulators
We conclude by discussing the bulk-edge correspondence, a principle in solid state physics, according to which one should be able to read the topology of the bulk physical system from the effects it induces on boundary states. This principle underlies, for example, the quantization of the Hall current on the boundary of a sample of a quantum Hall system.
In this section, we illustrate how Toeplitz extensions and the maps they induce in (bivariant) K-theory are essential for a mathematical understanding of these phenomena.

5.1.
The bulk-boundary correspondence for the one-dimensional Su-Schrieffer-Heeger model and the Noether-Gohberg-Krein index theorem. We will now give an exposition of the key ideas behind the bulk-edge correspondence for the one-dimensional Su-Schrieffer-Heeger model [39], a lattice model with chiral symmetry. Our main reference for this Subsection is [37, Chapter 1]. On the Hilbert space C 2 ⊗ C n ⊗ ℓ 2 (Z) we consider the one dimensional Hamiltonian (27) H := 1 2 (σ 1 + iσ 2 ) ⊗ 1 n ⊗ U + 1 2 (σ 1 − iσ 2 ) ⊗ 1 n ⊗ U * + mσ 2 ⊗ 1 n ⊗ 1, where 1 n and 1 are identity operators on C n and C 2 , respectively, m is a mass term, U is the right shift on ℓ 2 (Z) defined in (2), and the σ i are the Pauli matrices This Hamiltonian goes back to work of [39] and models a conducting polymer, namely polyacetilene. It possess a chiral symmetry, implemented by the unitary operator J = σ 3 ⊗ 1 n ⊗ 1, i.e., J * HJ = −H.
The model has a spectral gap at m = 0 so there exists ε > 0 and a continuous function 1 for x ∈ [0, ∞), so that we can form the Fermi projection P F := χ(H) through functional calculus with χ. The projection P F satisfies the identity JP F J = 1 − P F , so that the flat band Hamiltonian Q := 1 − 2P F = sgn(H) satisfies again J * QJ = −Q. Moreover, Q 2 = 1, hence its spectrum consists of the two isolated points +1 and −1, allowing us to write for U F a unitary on C n ⊗ ℓ 2 (Z). This unitary operator, called the Fermi unitary, provides us with a natural topological invariant for the boundary system, the first odd Chern number, which can be computed as follows. We use the discrete Fourier transform mentioned in (1) to write F QF * as a direct integral ⊕ S 1 Q z d z where each of the Q k 's has the form The family of unitary operators is differentiable and the first Chern class can be computed as the integral This quantity is an invariant under small perturbations.
Moreover, it has a spectral gap at 0 that we denote by ∆.
Let us now consider the Hilbert space obtained as the span of all the eigenvectors with eigenvalues in [−δ, δ] ⊂ ∆, which we denote by E δ . The chirality operator J can be diagonalised on E δ , and we have a splitting E δ = E δ + ⊕ E δ − . The difference of the dimensions of the spaces E δ ± is the boundary invariant of the system and it can be computed as a trace: tr( J P δ ) = N + − N − , N ± = dim E δ ± , where P δ := χ(| H| ≤ δ) is the spectral projection. This invariant is independend of the choice of δ, as long as it lies in the central gap.
The bulk-edge correspondence is contained in the following identity, that relates the bulk invariant (winding number of the Fermi unitary) to the boundary invariant we just introduced.
for the class of the extension (23), as constructed in [22] (see also [1]). It remains an interesting open question whether groupoid C * -algebras of higher dimensional systems admit a description in terms of C * -algebras associated to families of C * -correspondences, for instance in terms of product and subproduct systems [19,20,38,41].