An index theorem for quarter-plane Toeplitz operators via extended symbols and gapped Invariants related to corner states

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg-Krein theory for matrix factorizations, we extend the symbols defined originally on a two-dimensional torus to some three-dimensional sphere and derive a formula to express their Fredholm indices through extended symbols. Variants for families of (self-adjoint) Fredholm quarter-plane Toeplitz operators and those preserving real structures are also included. For some bulk-edge gapped single-particle Hamiltonians of finite hopping range on a discrete lattice with a codimension-two right angle corner, topological invariants related to corner states are provided through extensions of bulk Hamiltonians.


Introduction
Topological corner states took much interest in condensed matter physics as a characteristic of higher-order topological insulators. Aimed at studies of topological corner states, we discuss index theory for some Toeplitz operators on a discrete quarter-plane. Index theory for quarter-plane Toeplitz operators has been investigated by Simonenko, Douglas-Howe [32,14], and a necessary and sufficient condition for these operators to be Fredholm is obtained in terms of the invertibility of two associated half-plane Toeplitz operators. Index formulas for Fredholm quarterplane Toeplitz operators are obtained by Coburn-Douglas-Singer, Dudučava, Park [13,15,27]. Coburn-Douglas-Singer derived their formula by showing that there is a deformation to some quarter-plane Toeplitz operators of a standard form preserving Fredholm indices [13]. Dudučava employed Gohberg-Kreȋn theory for the factorization of some matrix functions on a circle [18,35,12] and obtained a formula by using a construction of parametrix [15]. Park obtained an index formula by a construction of a cyclic cocycle and using a pairing between K-theory and cyclic cohomology [27].
A characteristic feature of topological insulators is the existence of topological edge states. Although the bulk is gapped (insulating), edge states exist that account for the metallic properties of the boundary of the system. This appearance of edge states is known to originate from a topological invariant for the gapped bulk, called the bulk-edge correspondence. A typical example is the integer quantum Hall system in which the bulk topological invariant is known to be the first Chern number of a complex vector bundle (Bloch bundle) over a two-dimensional torus (Brillouin torus). Bellissard investigated the quantum Hall effect through noncommutative geometry [8], and Kellendonk-Richter-Schulz-Baldes gave a proof of the bulk-edge correspondence based on index theory for Toeplitz operators [24]. K-theory was employed for the classification of topological insulators [25,16] (see also [28] and the references therein). We note that matrix factorizations is also used in recent physical studies of (first-order) topological insulators [1]. For higher-order topological insulators [9,30], an actively studied topic in condensed matter physics, the bulk-edge correspondence is much generalized to include corner states. For two-dimensional second-order topological insulators, for example, the bulk and two edges whose intersection form a codimension-two corner are gapped, though there exist topological corner states, and a relation between some gapped topology and corner states are much discussed.
In [20], a mathematical approach to topological corner states is proposed based on index theory for quarter-plane Toeplitz operators, where topological invariants for bulk-edge gapped Hamiltonians are defined as elements of a K-group of some C * -algebra and its relation with hinge states is proved. Although this shows a relation between some gapped topology and corner states, gapped invariants are defined abstractly and much more geometric understanding is required, in order both for investigation from the physical point of view and its computation. For this purpose, we investigate further index theory for quarter-plane Toeplitz operators, especially Dudučava's idea of using matrix factorizations [15] from a topological point of view. We consider Fredholm quarter-plane Toeplitz operators of two-variable rational matrix function symbols. For each of them, there associates two invertible half-plane Toeplitz operators having the same symbol. In Sect. 3, we investigate the geometric implications of this invertibility condition. Through the Fourier transform in a direction parallel to the boundary, a half-plane Toeplitz operator corresponds to a one-parameter family of Toeplitz operators, and the problem reduces to a study of invertible Toeplitz operators. For an invertible Toeplitz operator of a rational matrix function symbol, Gohberg-Kreȋn theory states that there is a decomposition of the symbol as a product of two matrix-valued functions such that each factor of the decomposition can be analytically continued to a disk. By using analytic continuation, we see that the symbol of the quarter-plane Toeplitz operator defined originally on a two-dimensional torus can be extended as a continuous nonsingular matrix-valued function over some three sphere. This extension is shown to be independent of the choice of the factorization, therefore is canonically associated with our operator. We then show in Sect. 4 that the Fredholm index of the quarterplane Toeplitz operator is given through the three-dimensional winding number of the extended symbol (Corollary 4.10). Note that a part of its proof is based on Coburn-Douglas-Singer's idea [13]. Our formula can be extended to Fredholm quarter-plane Toeplitz operators which are self-adjoint, preserving real structures and families of them, and these variants are proved in a parallel way. In this paper, we mainly discuss families of (self-adjoint) Fredholm quarter-plane Toeplitz operators for which we use complex K-theory (Theorem 4.1), and the results for those operators preserving real structures are contained in Sect. 5. Necessary results about quarter-plane Toeplitz operators and Gohberg-Kreȋn theory for matrix factorizations used in this paper are collected in Sect. 2.
Applications to topological corner states are discussed in Sect. 6. We consider translation invariant single-particle Hamiltonians of finite hopping range on the lattice Z n in each of the ten Altland-Zirnbauer classes [2], and discuss its restrictions onto (Z ≥0 ) 2 × Z n−2 assuming the Dirichlet boundary condition. When the bulk Hamiltonian and its compressions onto two half-spaces Z × Z ≥0 × Z n−2 and Z ≥0 × Z × Z n−2 are gapped, we associates a nonsingular matrix function (over some three sphere for two-dimensional systems) through the matrix factorization which is an extension of the bulk Hamiltonian. We define a topological invariant for such a bulk-edge gapped Hamiltonian as a K-class in some topological K-theory group of this extended bulk Hamiltonian (Definition 6.1). A relation between this gapped topological invariant and corner/hinge states is given in Theorem 6.2, which provides a geometric formulation for the relation between the abstractly defined gapped topological invariant and corner states in [20]. In order to construct extensions of bulk Hamiltonians, we need to take matrix factorizations. For matrix factorizations of rational matrix functions on the unit circle in the complex plane, an algorithm is known [18,12,17] and the finite hopping range condition is assumed correspondingly. In [22], a classification of topological invariants related to corner states in each of the Altland-Zirnbauer classes are proposed based on index theory, where Boersema-Loring's formulation of KO-theory for real C * -algebras [10] is employed. Since topological corner states are one motivation of this work, some parts of the discussions in this paper are organized in this framework. For example, the real symmetries discussed in Sect. 5 are taken from Boersema-Loring's picture. Note that the integration formula for our gapped topological invariants, like the integration of the Berry curvature for the first Chern number, is still missing since our three sphere is not smooth, although our formulation provides a way to understand gapped topological invariants related to corner states in a geometric way. For example, that for a two-dimensional class AIII system is given by the three-dimensional winding number of the extension of the bulk Hamiltonian (Example 6.3) and that for a three-dimensional class A system is provided as a topological invariant for an extension of the Bloch bundle (Example 6.4).

Preliminaries
In this section, we collect the necessary results and notations used in this paper.
2.1. Quarter-Plane Toeplitz operators. Let T be the unit circle in the complex plane equipped with the normalized Haar measure. For f ∈ C(T n ), we write M f for the bounded linear operator on l 2 (Z n ) corresponding to the multiplication operator on L 2 (T n ) generated by f through the Fourier transform L 2 (T n ) ∼ = l 2 (Z n ). For an integer k, we write Z ≥k for the set of integers greater than or equal to k. Let δ n be the characteristic function of a point n ∈ Z and let l 2 (Z ≥0 ) be the closed subspace of l 2 (Z) spanned by {δ n | n ≥ 0}. Let P be the orthogonal projection of l 2 (Z) onto l 2 (Z ≥0 ). For f ∈ C(T), the operator on l 2 (Z ≥0 ) defined by T f ϕ = P M f ϕ for ϕ ∈ l 2 (Z ≥0 ) is called the Toeplitz operator of continuous symbol f . Let T be the C * -algebra generated by those Toeplitz operators. We have the following Toeplitz extension: where σ is a * -homomorphism that maps T f to its symbol f . Let δ m,n be the characteristic function of the point (m, n) in Z 2 , and let H 0 , H ∞ and H 0,∞ be closed subspaces of l 2 (Z 2 ) spanned by {δ m,n | n ≥ 0}, {δ m,n | m ≥ 0}, and {δ m,n | m ≥ 0 and n ≥ 0}, respectively. Let P 0 , P ∞ and P 0,∞ be the orthogonal projection of l 2 (Z 2 ) onto H 0 , H ∞ and H 0,∞ , respectively. Note that respectively, are called half-plane Toeplitz operators. The operator on H 0,∞ defined by T 0,∞ f ϕ = P 0,∞ M f ϕ for ϕ ∈ H 0,∞ is called the quarter-plane Toeplitz operator. Let T 0 and T ∞ be the C * -algebras generated by half-plane Toeplitz operators of the form T 0 f and T ∞ f , respectively, and let T 0,∞ be the C *algebra generated by quarter-plane Toeplitz operators. Note that T 0 ∼ = C(T) ⊗ T and T ∞ ∼ = T ⊗C(T) by the Fourier transform in a direction parallel to the boundary of half-planes. Corresponding to these isomorphisms, let σ 0 = 1 C(T) ⊗ σ and σ ∞ = σ ⊗ 1 C(T) which are * -homomorphisms from T 0 and T ∞ to C(T 2 ). Let S 0,∞ be the pullback C * -algebra of these two * -homomorphisms, The following short exact sequence of C * -algebras is known [27]: where K is the compact operator algebra, the map K → T 0,∞ is the inclusion, and γ is a * -homomorphism that maps T 0,∞

Matrix Factorizations.
In this subsection, we collect necessary results about Gohberg-Kreȋn theory for factorizations of rational matrix functions in cases on the unit circle in the complex plane. For details, we refer the reader to [18,12,17]. Let us consider the Riemann sphereĈ = C ∪ {∞}. The unit circle T is contained inĈ andĈ \ T consists of two connected components. Let D + = {z ∈ C | |z|< 1} and D − = {z ∈ C | |z|> 1} ∪ {∞} which are open disks. We write D 2 for the closed unit disk T ∪ D + in the complex plane. Let f : T → GL n (C) be a nonsingular rational matrix function, that is, a nonsingular matrix function of entries consisting of rational functions with poles off T. We have the following decomposition, called a (right) matrix factorization: where f − , Λ and f + are rational matrix functions on T satisfying the conditions below.
• f + (resp. f − ) admits a continuous extension onto T ∪ D + = D 2 (resp. T ∪ D − ) as a nonsingular matrix function and is analytic on D + (resp. D − ). We write f e + (resp. f e − ) for the extension. • Λ is the diagonal matrix function of the form Λ(z) = diag(z κ1 , . . . , z κn ) with nonincreasing sequence of integers κ 1 ≥ · · · ≥ κ n called partial indices. Among the many results known for matrix factorizations, we note the following: • Remark 2.1. There is known a general class of Banach algebras of functions on the circle that admits matrix factorizations called decomposing Banach algebras [12]. One example is the Wiener algebra over the circle consisting of all complexvalued functions f on T admitting an absolutely convergent Fourier series. The results in this paper are also valid for such continuous matrix functions admitting matrix factorizations. Note that the algebra C(T) of continuous functions is not decomposing. In this paper, we mainly discuss rational matrix functions on the unit circle in view of our applications discussed in Sect 6.

Extension of Symbols Through Matrix Factorizations
Let f : T 2 → GL n (C) be a continuous map. The associated quarter-plane Toeplitz operator T 0,∞ f is Fredholm if and only if the half-plane Toeplitz operators T 0 f and T ∞ f are invertible [14]. In this section, we discuss the geometric implication of this condition when the symbol f is a rational matrix function for both of the two variables of T 2 . By using matrix factorizations, we provide a way to extend f to a nonsingular matrix-valued continuous function on a three sphere (Sect. 3.1). We also discuss its variants for families of them (Sect. 3.2) and the cases when matrix functions take value in (skew-)hermitian matrices (Sect. 3.3).

Extension of Nonsingular Matrix Functions of Trivial Partial Indices.
Let f be a nonsingular rational matrix function on the circle T, and assume that the associated Toeplitz operator T f is invertible. Let f = f − f + be a canonical factorization of f . Let f e + and f e − be the associated extensions of f + and f − onto T ∪ D + and T ∪ D − , respectively. For two disks D 2 = T ∪ D + and T ∪ D − , we consider the following identification: where we set I(0) = ∞. By using this identification, we associate the following nonsingular matrix-valued continuous map for a canonical factorization: f e coincides with f on T and is its continuous extension onto D 2 .
By Lemma 3.1, for a rational matrix function f : T → GL n (C) whose associated Toeplitz operator T f is invertible, there is a canonically associated extension f e onto the disk D 2 . For a non-negative real number t, we write T t = {z ∈ C | |z| = t}. For 0 ≤ t ≤ 1, let m t : T → T t be the map defined by m t (z) = tz. We take the pullback m * t (f e | Tt ) of f e | Tt onto T by m t . Let us consider the Toeplitz operator T t := T m * t (f e | T t ) associated with this matrix function. In other words, we consider D 2 to be a family of circles of radius 0 ≤ t ≤ 1 and consider the associated family of Toeplitz operators. For these operators, the following holds: Lemma 3.2. Let f be a nonsingular rational matrix function on T of trivial partial indices, and consider its extension f e : D 2 → GL n (C) in (4). For 0 ≤ t ≤ 1, the Toeplitz operator T t in is invertible.
Proof. The invertibility of T 1 = T f is included in our assumption. When t = 0, the Toeplitz operator T 0 is associated with nonsingular constant matrix function and is invertible. Let us consider the case in which 0 < t < 1. For z ∈ T, we have . Therefore, we have the following decomposition for the symbol of T t : . Each component of equation (5) is a rational matrix function on T. Let D +,t = {z ∈ C | |z| < t} and D −,t −1 = {z ∈ C | |z| > t −1 } ∪ {∞}. Let us consider the maps m +,t : D + → D +,t , z → tz, and m −,t −1 : and f e + are extensions of f − and f + that are analytic on D − and D + , pullbacks of their restrictions m * −,t −1 (f e − | D −,t −1 ) and m * +,t (f e + | D+,t ) onto D − and D + also provide such extensions of m * t −1 (f e − | T t −1 ) and m * t (f e + | Tt ). Therefore, equation (5) is a canonical factorization of m * t (f e | Tt ), and the associated Toeplitz operator T t is invertible.

3.2.
Extension of Families of Matrix Functions. We next extend the discussions in Sect. 3.1 to families of rational matrix functions of trivial partial indices. Matrix factorizations for such families are studied byŠubin [35]. Lemma 3.3. Let X be a topological space. Let f : T×X → GL n (C) be a continuous map such that for each x ∈ X, f (x) is a rational matrix function on T of trivial partial indices. Through the matrix factorization, there canonically associates a continuous map f e : D 2 × X → GL n (C) that extends f . Proof. Following [35], for each x 0 ∈ X, there exists an open neighborhood U ⊂ X of x 0 and continuous matrix functions is a canonical factorization. By using this factorization, we obtain a continuous extension f e : D 2 × U → GL n (C) of f as in equation (4). As in Lemma 3.1, this f e is independent of the choice of canonical factorization. We cover X by such open sets {U α } α∈J . When U α ∩ U β = ∅, we may consider two extensions of f | T×(Uα∩U β ) onto D 2 × (U α ∩ U β ) corresponding to extensions onto D 2 × U α and D 2 × U β , and they coincide by Lemma 3.1. Therefore, we obtain the desired extension onto D 2 × X.
We next consider families of two-variable rational matrix functions whose associated quarter-plane Toeplitz operators are Fredholm. Let which is topologically a three-dimensional sphere. Since D 2 ⊂ C, we considerS 3 as a subspace of C 2 and use complex variables (z, w) ∈ C 2 to parametrizeS 3 . Proposition 3.4. Let X be a topological space. Let f : T 2 × X → GL n (C) be a continuous map such that for each x ∈ X, f (x) is a two-variable rational matrix function for which the associated quarter-plane Toeplitz operator T 0,∞ f (x) is Fredholm. Through matrix factorization, there canonically associates a continuous map [14]. Through a Fourier transform in a direction parallel to the boundary, the invertible half-plane Toeplitz operator T 0 f (x) corresponds to a family of invertible Toeplitz operators {T f (z,·,x) } z∈T parametrized by the circle. Therefore, by Lemma 3.3, there canonically associates a continuous extension f e : we also obtain an extension of f onto D 2 × T × X through matrix factorization. Combined with them, we obtain an extension f E of f as a nonsingular matrix-valued continuous function onS 3 × X.

Hermitian and Skew-Hermitian Matrix Functions.
In this subsection, we discuss the case in which the nonsingular rational matrix functions in Sect. 3.1 and 3.2 take values in hermitian or skew-hermitian matrices. Let GL n (C) sa (resp. GL n (C) sk ) be the space of n-by-n hermitian (resp. skew-hermitian) invertible matrices.
Lemma 3.5. Let I be GL n (C) sa or GL n (C) sk . Let X be a topological space and f : T × X → I be a continuous map such that, for each x ∈ X, f (x) is a rational matrix function on T with trivial partial indices. Then the extension f e of f in Lemma 3.3 is also a hermitian or skew-hermitian matrix function; that is, f e : D 2 × X → I.
Proof. We consider hermitian matrix functions, that is, the case in which I = GL n (C) sa . By Lemma 3.3, it is sufficient to show that f e (x) for each x ∈ X is a hermitian matrix-valued function. Therefore, it is sufficient to consider the case when X is one point set and we assume this condition.
The uniqueness of analytic continuation leads to the . The result for skew-hermitian matrix functions is proved in a similar manner.
By Proposition 3.4 and Lemma 3.5, we obtain the following result: is Fredholm. Through matrix factorization, there canonically associates a continuous map f E :S 3 × X → I that extends f .

Index Theorem for Quarter-Plane Toeplitz Operators via Extended Symbols
In this section, we give a formula to express family indices for (self-adjoint) Fredholm quarter-plane Toeplitz operators of two-variable rational matrix function symbols by using extended symbols obtained through the matrix factorizations in Sect. 3. The main theorem in this section is Theorem 4.1 which is formulated by using complex K-theory and we start from preliminaries of K-theory.
Let n 0 = 2 and n 1 = 1. Let M (0) n (C) = M 2n (C) sa , that is, the set of 2nby-2n hermitian matrices, and M (1) where the equivalence relation ∼ i is generated by homotopy and stabilization by I (i) . For a compact Hausdorff space X, complex topological K-groups are defined as K −i (X) = K i (C(X)). For a locally compact Hausdorff space Y , we denote K −i cpt (Y ) for the compactly supported K-group of Y . For the basics of K-theory used in this paper, we refer the reader to [4,5,7,23,29,10]. Let i = 0 or 1, and let X be a compact Hausdorff space. Let f : T 2 × X → GL (i) N (C) be a continuous map, such that for each x ∈ X, f (x) is a two-variable rational matrix function, and the associated quarter-plane Toeplitz operator } x∈X is a family of Fredholm operators which defines an element of the even complex K-group K 0 (X). When i = 0, {T 0,∞ f (x) } x∈X is a family of self-adjoint Fredholm operators. In this case, we may stabilize f if necessary (i.e., take a direct sum with I (0) ) and assume that the essential spectrum is not contained in the set of positive real numbers R >0 or the set of negative real numbers R <0 for any x ∈ X. Under this assumption, {T 0,∞ f (x) } x∈X defines an element of the odd complex K-group K 1 (X). In both of these cases, we write [T 0,∞ f ] for the K-class of the family of (self-adjoint) Fredholm 1 quarterplane Toeplitz operators. By using matrix factorization, there is an associated continuous extension f E of f ontoS 3 × X that takes values in hermitian invertible matrices (when i = 0; see Proposition 3.6) or invertible matrices (when i = 1; see Let us consider the following isomorphism, We write β : through the above decomposition and the Bott periodicity iso- The following is the main theorem in this paper. be a continuous map such that for each x ∈ X, f (x) is a two-variable rational matrix function, and the associated quarter-plane Toeplitz operator be the extension of f through matrix factorization in the hermitian case of Proposition 3.6 (when i = 0) or Proposition 3.4 (when i = 1).
For the f in Theorem 4.1, the associated family of half-plane Toeplitz opera- be the boundary map of the six-term exact sequence for K-theory of C * -algebras associated with the following extension obtained by taking a tensor product of the sequence (2) with C(X), 1 Self-adjoint Fredholm operators when i = 0 and Fredholm operators when i = 1. In this section, the term (self-adjoint) should be read when i = 0.
]. Let us consider the following diagram: where ∂ T is the boundary map associated with the Toeplitz extension (1) and ∂ pair is the boundary map of the six-term exact sequence of topological K-theory associated with the pair ( Maps ∂ T ⊕ −∂ T and ∂ pair are surjective, and the kernel of ∂ pair is the diagonals that are contained in the kernel of ∂ T ⊕ −∂ T . Therefore, the horizontal map α making the diagram commutative is induced. To show Theorem 4.1, we will construct the following key diagram: The map α is surjective and the induced homomorphismᾱ in the above diagram is an isomorphism. Note that there is the following isomorphism, Through this decomposition, Ker(α) ∼ = K −i (X) and we have the isomorphism, Through this isomorphism, the mapᾱ correspond to the Bott periodicity isomor-

4.1.
Construction of the Map ψ. In this subsection, we construct the homomorphism ψ in the diagram (9). For its construction, Coburn-Douglas-Singer's idea to use homotopy lifting property for some fibrations plays a key role, and we introduce them first. Let U n (T ) and U n (C(T)) be subspaces of M n (T ) and M n (C(T)), respectively, consisting of unitary elements, let U n (T ) sa and U n (C(T)) sa be subspaces of selfadjoint unitaries. The map σ in (1) induces the following maps, which we also denote as σ: (11) σ : U n (T ) → U n (C(T)).
The map (11) is a Hurewicz fibration, which is used in [13]. We also have its variants (Lemma 4.2 and Proposition 5.5) which will be well-known [37,7], though we briefly contain its proof since, as in [13], they will play a key role in our discussion. The proof of Lemma 4.2 is simply an application of discussions around Proposition 4.1 of [7] to the Toeplitz extension. For a space Y and its element y, we write Y y for the connected component of Y containing y. For a Banach algebra A, its subset S ⊂ A and an R-linear operator s on A of order two, we write S s and S −s for the subsets of S that are pointwise fixed by s and −s, respectively.
Proof. Since U n (C(T)) sa is a paracompact Hausdorff space, it is sufficient to show that (12) is a fiber bundle [33]. Let u be a self-adjoint unitary in M n (C(T)). There exists a self-adjoint invertible lift x ∈ M n (T ) of u since a self-adjoint Fredholm operator can be perturbed to a self-adjoint invertible operator by a self-adjoint compact operator. Then, the self-adjoint unitary x|x| −1 ∈ M n (T ) satisfies σ(x|x| −1 ) = u, and the map σ : U n (T ) sa → U n (C(T)) sa is surjective. It is now sufficient to show that, for any s ∈ U n (T ) sa , the map σ : U n (T ) sa s → U n (C(T)) sa σ(s) is a fiber bundle. Note that the map σ : M n (T ) → M n (C(T)) has a continuous linear section given by compression, that is, mapping f to T f . Let k = s − T σ(s) which is a self-adjoint compact operator, and let l : M n (C(T)) → M n (T ) be a map given by l(f ) = T f + k. This map l preserves self-adjoint elements and satisfies l(σ(s)) = s. Combined with the map d : GL n (T ) → U n (T ), d(T ) = T |T | −1 , we obtain a continuous local section of the map σ : U n (T ) sa s → U n (C(T)) sa σ(s) in a neighborhood of σ(s) that maps σ(s) to s. Let s be an operator on M n (T ) given by the conjugation of s. Then, as in [37], the quotient map GL n (T ) 1 → (GL n (T )/(GL n (T )) s ) 1 has a continuous local section which follows from [26]. Combined with the map d, we obtain a continuous local section of the quotient map U n (T ) 1 → (U n (T )/U n (T ) s ) 1 . Let us consider the map U n (T ) → U n (T ) sa given by T → T sT * , that comes from the action of unitaries to self-adjoint unitaries by conjugation. Its stabilizer subgroup at s is U n (T ) s . As in Lemma 4.1 of [37], each orbit of this action is open and induces a homeomorphism (U n (T )/U n (T ) s ) 1 ∼ = −→ U n (T ) sa s . Let t be an operator on M n (C(T)) given by the conjugation of σ(s). We consider similar discussion for the algebra C(T) and obtain the following diagram, whereσ is the map induced by σ. Therefore, U n (T ) 1 → (U n (C(T))/U n (C(T)) t ) 1 has a continuous local section and the result follows (see Sect. 7.4 of [34]).
) satisfying the following conditions: as an operator on (iv) There exist L ∈ N and a continuous map as an operator on ) as a family of Toeplitz operators parametrized by T × X, and similarly for (iv).
Proof. We first take the deformation retraction of invertible elements T 0 and T ∞ to unitaries; that is, for . We next consider the following two homomorphisms, Note that K i (S 0,∞ ) = Z for i = 0, 1, and that the map (σ S ⊗ 1 C(X) ) * is computed by using the Künneth theorem [31,27]. The images of the above two maps coincide.
. Note that π * is injective, and the K-class [g] of g is uniquely determined. Since the equivalence relation used to define the K-group K −i (T 2 × X) is generated by homotopy and stabilization, if we take N ′ to be sufficiently large, there also exists a homotopy in C(T 2 × X, U (i) N ′ −N to π * g. By the homotopy lifting property of fibration (11), there exists a homotopy T 0 is independent of z ∈ T. By a similar discussion for T ∞ , we obtain a homotopy , we obtain the following: We perturb k 0 and k ∞ to families of (self-adjoint) finite rank operators to obtain homotopies T 0 t and T ∞ t for 2 3 ≤ t ≤ 1 in the space of (selfadjoint) invertible half-plane Toeplitz operators and obtain the desired result.
For simplicity, in what follows, we assume that our representative (T 0 , T ∞ ) of the K-class [(T 0 , T ∞ )] in K i (S 0,∞ ⊗C(X)) is sufficiently stabilized to take N = N ′ in Lemma 4.3. Under this assumption, we take a continuous map g, a homotopy {g t } 0≤t≤1 from f to π * g in C(T 2 × X, GL  (14). For these elements, the following holds [13]:
Let {a t } 0≤t≤1 (resp. {b t } 0≤t≤1 ) be a path from t 0 (resp.t ∞ ) to K (resp. L) g ⊕ · · · ⊕ g in C(T×X, M is (self-adjoint) invertible and independent of z ∈ T. F is a family of (self-adjoint) Fredholm operators on [−2, 2] × T × X, which is invertible on ([−2, −1] ⊔ [1, 2]) × T × X, therefore, it defines the following element of the relative K-group, We write for the class of [F ] in the quotient group, and we define

Construction of the Map φ.
In this subsection, we construct the following group homomorphism φ in the diagram (9).
, and the image of this map is the same as that of π * in (13). Therefore, as in the proof of Lemma 4.3, there exists N ′ ≥ N , a continuous map g : X → GL (i) N ′ (C) and a homotopy {g t } 0≤t≤1 from g 0 = f r ⊕ I (i) N ′ (C)). As in Sect. 4.1, we assume that the representative f of the K-class [f ] is sufficiently stabilized to take N ′ = N for simplicity.
Let R : T 2 → T 2 be an orientation-preserving homeomorphism given by R(z, w) = (w,z). Corresponding to the notation (14) for half-plane Toeplitz operators, we writek = (R × id X ) * k for a matrix-valued continuous function k on T 2 × X; that is k(z, w, x) = k(w,z, x). By using this notation, let G : [−2, 2] × T 2 × X → GL (i) N (C) be a continuous map defined as follows: Since g 1 (z, w, x) = g(x) does not depend on z and w, this family G defines a continuous map on [−2, 2] × T 2 × X. We consider a family of Toeplitz operators T G by using second T parametrized by w for the symbols of Toeplitz operators; that is, for (s, z, x) ∈ [−2, 2] × T × X, we set T G is a family of Toeplitz operators of invertible symbols and therefore a family of Fredholm operators parametrized by [−2, 2] × T × X. When s = ±2, these Toeplitz operators have constant invertible symbols and are invertible. This family Proposition 4.6. φ is a well-defined group homomorphism.

Proof. We show that [[T G ]] depends only on the K-class
The proof is divided into three steps.
(i) We first fix f and g and show that [[T G ]] is independent of the choice of homotopy g t . For this purpose, we take another homotopy {h t } 0≤t≤1 from h 0 = f r to h 1 = π * g in C(T 2 × X, GL  In what follows, we show that (16) [ Let us consider the following map, which is constructed as follows: • For spaces, two endpoints of two intervals [−2, 2] are connected (connect ±2 with ∓2) to construct the circle T. • For two families of Fredholm operators that are invertible on {±2} × T × X, we connect invertible operators on two endpoints correspondingly by a continuous path of (self-adjoint) invertible families. This is possible and unique up to homotopy by Kuiper's theorem. • We then obtain a family of Fredholm operators on T 2 × X, which defines an element of K −i+1 (T 2 × X). Note that we have the following isomorphism: be the inclusion corresponding to the direct sum decomposition in (17). The map G is the same as the composite of the addition of the group K −i+1 (([−2, 2], {±2}) × T × X) followed by ι. Let us consider the projection ̟ : K −i+1 (T 2 ×X) → K −i−1 (X) corresponding to the decomposition in (17). Through the identification (10), [[T G ]] and ̟ • ι([T G ]) corresponds. Therefore, in order to show (16), it is sufficient to show that Let us discuss the element G([T G ], [T H ′ ]). The families of Toeplitz operators T G and T H ′ have symbols G and H ′ , respectively, which are shown in Figure 2. Since these two have the same matrix functions at the boundaries, we can glue G and H ′ as indicated in Figure 2, and obtain a continuous map T 3 × X → GL   2] is consistent with that for each homotopy (e.g., g t ) or not (e.g., h −t ) that defines an element of the group  Figure 2, there are the same matrix functions with opposite parameter directions; therefore these dashed areas can be cancelled by a homotopy, and this K-class is the same as the K-class of the continuous map a : As shown in Figure 3, Let us consider the homeomorphism q × R on [−1, 1]/{±1} × T 2 given by (q × R)(s, z, w) = (−s, w,z), and consider the map (q × R × id X ) * on the K-group holds. Furthermore, corresponding to the direct sum decomposition (19) K the map (q × R × id X ) * acts on the direct summand K −i−3 (X) as multiplication by −1. By the map ∂ T , this direct summand K −i−3 (X) in (19) maps isomorphically to the direct summand K −i−1 (X) in (17). Therefore, we have ̟( , which leads to equation (18); that is, (ii) We next fix f and show that [[T G ]] is independent of the choice of a representative g of the K-class [g] in K −i (X) satisfying [f r ] = π * [g]. Note that the K-class [g] is uniquely determined since π * is injective. Let g ′ : X → GL (i) N ′ (C) be another representative; that is, [g] = [g ′ ]. By taking N and N ′ sufficiently large, we assume that N = N ′ and that there exists a homotopy {k t } 0≤t≤1 from k 0 = g to k 1 = g ′ in C(X, GL (i) N (C)). Combined with the homotopies {g t } 0≤t≤1 (from f r to π * g) and {π * k t } 0≤t≤1 , we obtain a homotopy from f r to π * g ′ in C(T 2 × X, GL Within − 1 2 ≤ s ≤ 1 2 , this family K has k t of the opposite parameter direction, and it can be collapsed by a homotopy to obtain G. This homotopy preserves s = ±2 and provides a homotopy from T G to T K in the space of families of (self-adjoint) Fredholm operators that are invertible at s = ±2; therefore, . Combined with the homotopy {g t } 0≤t≤1 , we obtain a homotopy from f r to π * g, which provides the family of Toeplitz operators T G ′ and a homotopy from T G to T G ′ . Explicitly, for 0 ≤ u ≤ 1, let G u : [−2, 2] × T 2 × X → GL Then, the associated family of Toeplitz operators {T Gu } u∈[0,1] provides a path of families of (self-adjoint) Fredholm operators invertible at s = ±2 satisfying T G0 = T G and T G1 = T G ′ ; therefore, The following lemma states that the diagram (9) commutes: Proof. The map β is given through the decomposition (6) and the square of the Bott periodicity isomorphism. The mapᾱ corresponds to the Bott periodicity isomorphism, and we see a relation between the map φ and the Bott periodicity. Let π ′ :S 3 ×X → X be the projection. Through the decomposition (6), the direct summand K −i (X) in K −i (S 3 ×X) is represented by [π ′ * g], where g : X → GL (i) N (C) for some N . π ′ * g is a continuous map onS 3 × X, which is constant with respect toS 3 . In this case, a homotopy {g t } 0≤t≤1 to construct G in (15) can be taken to be constant g t = π * g, and the corresponding family T G of Toeplitz operators representing φ([π ′ * g]) to be a family of (self-adjoint) invertible operators. Therefore, φ([π ′ * g]) = 0, and the direct summand K −i (X) maps to zero through φ.
We next consider the direct summand of K −i (S 3 ×X) corresponding to K −i cpt (R 3 × X) in (6). We considerS 3 as the one-point compactification of R 3 , where the point at infinity corresponds to the base point s 0 = (1, 1) ∈S 3 . K-classes in this component are represented by a continuous maps f : N for any x ∈ X. SinceS 3 \ {s 0 } ∼ = R 3 is contractible, we can take a small three-dimensional ball B inS 3 and deform f to a continuous map which is I N , we can take a homotopy {g t } 0≤t≤1 to construct G in (15) as g t = I is simply the push-forward map through the Toeplitz operators [5]. Explicitly, it is the same as the composite of the following maps: In the above, the map from K −i cpt (R 3 × X) to K −i cpt (R × T 2 × X) is constructed as follows: we embed R 3 into R × T 2 as a small open ball. A K-class in K −i cpt (R 3 × X) is represented by a continuous map g from R 3 × X to GL N . Therefore, φ on K −i cpt (R 3 ×X) is given by the Bott periodicity isomorphism [5].
In summary, the compositeᾱ • φ is a projection onto K −i cpt (R 3 × X) followed by the square of the Bott periodicity isomorphism, which is β.     Figure 4). Note that T 0 f is identified with T f through the isomorphism (a)

Quarter-Plane Toeplitz Operators Preserving Real Structures
In this section, we discuss a variant of Theorem 4.1 in real K-theory. For this purpose, we use Atiyah's KR-theory for spaces equipped with involutions [3] and Boersema-Loring's formulation for the KO-theory of real C * -algebras [10]. Index theory for quarter-plane Toeplitz operators preserving some real structures and application to topological corner states are discussed in [22], which we mainly follow.

Matrix Functions Preserving Real or Quaternionic
Structures. Let C be a real or a quaternionic structure on C n , that is, an antiunitary operator on C n whose square is +1 or −1. Note that, when we consider a quaternionic structure, the positive integer n must be even. We write Ad C for a real linear automorphism of order two on M n (C) given by Ad C (x) = CxC * . We also write * for the operation on M n (C) taking the hermitian conjugate of matrices. We write c for complex conjugation on C, that is, c(z) =z. Then, (T, c) is a Z 2 -space, which is a Z 2 -subspace of (D 2 , c).
Let (X, ζ) be a Z 2 -space and f : (T × X, c × ζ) → I be a Z 2 -equivariant continuous map such that for each x ∈ X, f (x) is a rational matrix function of trivial partial indices. Then, its extension f e in Lemma 3.3 is a Z 2 -map f e : (D 2 × X, c × ζ) → I.
Proof. By Lemma 3.3, the map f e is continuous. We now show its Z 2 -equivalence.
For each x ∈ X, we take a canonical factorization f (x) = f − (x)f + (x).
(i) We first consider the case in which I = (GL n (C), Ad C ). By assumption, the following equation holds for (z, x) ∈ T × X: Let f e + (x) and f e − (x) be continuous extensions of f + (x) and f − (x) onto T ∪ D + and T ∪ D − , which are analytic on D + and D − , respectively. The function z → Cf + (z, ζ(x))C * is a rational matrix function on T and Cf e + (z, ζ(x))C * provides its continuous extension onto T ∪ D + which is analytic on D + as a nonsingular matrix function. A similar observation holds for Cf − (z, ζ(x))C * and the equation (22) provides two canonical factorizations of f (x). Therefore, for each x ∈ X, there exists B ∈ GL n (C) such that Bf + (z, x) = Cf + (z, ζ(x))C * and f − (z, x)B −1 = Cf − (z, ζ(x))C * for z ∈ T. By the uniqueness of analytic continuation, we obtain )C * · Cf e + (z, ζ(x))C * = Cf e (z, ζ(x))C * . (ii) We next consider the case of I = (GL n (C), Ad C • * ). By assumption, we have for (z, x) ∈ T×X. The matrix function Cf e + (z −1 , ζ(x)) * C * (resp. Cf e − (z −1 , ζ(x)) * C * ) provides a continuous extension of Cf + (z, ζ(x)) * C * (resp. Cf − (z, ζ(x) * C * ) onto T ∪ D − (resp. T ∪ D + ), which is analytic on D − (resp. D + ), and the right hand side of equation (23) is also a canonical factorization of f . Therefore, as in the proof of Lemma 5.1, there exists B ∈ GL n (C) satisfying, Let ν be an involution 3 onS 3 given by the restriction of c 2 = c × c on C 2 ontõ S 3 . By Proposition 3.4 and Lemmas 5.1 and 5.2, we obtain the following result. Proposition 5.3. Let (X, ζ) be a Z 2 -space, and let I be a Z 2 -space (GL n (C), Ad C ), (GL n (C), Ad C • * ), (GL n (C) sa , Ad C ) or (GL n (C) sa , − Ad C ). Let f : (T 2 × X, c 2 × ζ) → I be a Z 2 -continuous map such that for each x ∈ X, f (x) is a two-variable rational matrix function for which the associated quarter-plane Toeplitz operator T 0,∞ f (x) is Fredholm. Then, through matrix factorization, there canonically associates a Z 2 -continuous map f E : (S 3 × X, ν × ζ) → I that extends f .

5.2.
Index Theorem: Real Cases. Let R be the antiunitary operator on C n given by R = diag(c, · · · , c), where c is the complex conjugation on C. Let j be an antiunitary operator on C 2 given by j(x, y) = (−ȳ,x). When n is even, let J = diag(j, · · · , j) be a quaternionic structure on C n . Let A be a unital C * -algebra equipped with a real structure 4 r. Let τ be the antiautomorphism on A of order two given by τ (a) = r(a * ). We call τ the transposition and write a τ for τ (a). The pair (A, τ ) is called C * ,τ -algebra in [10]. The transposition τ on A is extended to the transposition on the matrix algebra M n (A) by (a ij ) τ = (a τ ji ). Let ♯ ⊗ τ be a transposition on M 2 (A) defined by  Table 1. Let GL (i) n (A, τ ) be the set of all 3 The Z 2 -space (S 3 , ν) is Z 2 -equivariantly homeomorphic to the Z 2 -space S 2,2 in [3]. 4 An antilinear * -automorphism on A satisfying r 2 = 1.
be the boundary map of the 24-term exact sequence for KO-theory associated with the short exact sequence of C * ,τ -algebras, ]. For (S 3 , ν), we take a Z 2 -fixed point s 0 = (1, 1) inS 3 as its base point, and obtain an isomorphism ζ) be the projection corresponding to this decomposition.
r,N (C) be a Z 2 -map such that for each x ∈ X, f (x) is a two-variable rational matrix function and the associated quarter-plane Toeplitz operator The proof of Theorem 4.1 concerns three parts: matrix factorizations, the homotopy lifting property and K-theory. For Theorem 5.4, matrix factorizations are discussed in Sect. 5.1. Here, we note the following result concerning the Z 2equivariant homotopy lifting property. For C = R or J , let c T be an involution on M n (T ) given by c T (T ) = CT C * , and let c T be an involution on M n (C(T)) given by c T (f )(z) = Cf (z)C * for z ∈ T. Since (11) and (12) are Hurewicz fibrations and by Theorem 4.1 of [11], to show Proposition 5.5, it is sufficient to show that the restrictions on the Z 2 -fixed point sets are fibrations.
For (ii), we put unitaries preserving these real structures in the framework of Wood [37]. Let A = M 2n (T ) and set U (A) = U 2n (T ). Let and letc T : A → A be a real linear automorphism of order two given byc T (a) = CaC * . Its fixed point set Ac T is a unital real Banach * -algebra containing e. Let e be an operator on Ac T given by the conjugation of e. Let us consider the space (U (Ac T ) sk ) −e , that is, the elements in A = M 2n (T ) which are skew-adjoint unitary, commute withC and anti-commute with e. We have the following identification, Let u ∈ U n (T ) cT • * and s = 0 −u * u 0 ∈ (U (Ac T ) sk ) −e . We consider an operator s on U (Ac T ) e given by the conjugation of k. By Lemma 4.2 of [37], we have the following homeomorphism, given by [T ] → T sT * . We also have a similar homeomorphism for the algebra C(T), and as in the proof of Lemma 4.2, we obtain that the map We consider a translation invariant Hamiltonian on the lattice Z n of the following form: where a j1···jn ∈ M N (C), S j is the shift operator in the j-th direction and the subscript finite means that a j1···jn = 0 except for finitely many (j 1 , . . . , j n ) ∈ Z n . We assume that H is self-adjoint and consider such Hamiltonians in each of the ten Altland-Zirnbauer classes. In classes A, AI and AII, we further assume that the spectrum of the bulk Hamiltonian is not contained in R >0 and R <0 . Through the Fourier transform, the bulk Hamiltonian corresponds to a hermitian matrix-valued function H : T n → M N (C) sa on the n-dimensional Brillouin torus T n . We write (z 1 , . . . , z n ) for an element in T n . Corresponding to our finite hopping range condition, each entry of this matrix consists of a n-variable Laurent polynomial; therefore, the bulk Hamiltonian H correspond to a n-variable rational matrix function on T n . We next introduce our models for two edges and the corner. Note that, for each z = (z 3 , . . . , z n ) ∈ T n−2 , the matrix function H(z) on T 2 is a two-variable rational matrix function. Let H 0 (z) = T 0 H(z) and H ∞ (z) = T ∞ H(z) be the associated halfplane Toeplitz operators, and let H 0,∞ (z) = T 0,∞ H(z) be the associated quarter-plane Toeplitz operator. That is, for models of two edges, we consider the restrictions of our bulk Hamiltonian onto half-spaces Z × Z ≥0 × Z n−2 and Z ≥0 × Z × Z n−2 , and for the model of codimension-two right angle corner, we consider the restriction onto the lattice (Z ≥0 ) 2 × Z n−2 , where we assume the Dirichlet boundary condition. We assume that, for any z ∈ T n−2 , half-plane Toeplitz operators H 0 (z) and H ∞ (z) are invertible. Under this assumption, H(z) is also invertible and H 0,∞ (z) is Fredholm. Therefore, we assume that our model Hamiltonians for the bulk and two edges that makes the corner are gapped. Under this assumption, we discuss a relation between some gapped topological invariant and corner states. As in [22], the family of self-adjoint Fredholm operators {H 0,∞ (z)} z∈T n−2 defines an element of the complex K-group K −i+1 (T n−2 ) for classes A and AIII, or the KR-group KR −i+1 (T n−2 , c n−2 ) for classes AI, BDI, D, DIII, AII, CII, C and CI of some degree i corresponding to its Altland-Zirnbauer class ♠ as indicated in Table 2. We write I ♠ Gapless (H) for this element of the K-group. If I ♠ Gapless (H) is non-trivial, there exist topological corner/hinge states. For classes AIII, BDI, DIII, CII and CI where the Hamiltonians preserve chiral symmetry, a Hamiltonian H anti-commute pointwise with the chiral symmetry operator Π and can be represented by the off-diagonal form H = 0 h * h 0 . This h is also a nonsingular n-variable rational matrix function on T n . As in [22], this H or h preserves the symmetries of Boersema-Loring's formulation of complex or real K-theory groups. Under our assumption, by using matrix factorizations (Proposition 3.6 and Proposition 5.3), H or h on T n is extended to a nonsingular matrix-valued continuous Theorem 6.2 provides a geometric formulation for a relation between a gapped topological invariant and corner states in [20,22], though, since our three spherẽ S 3 is not smooth as the boundary of D 2 × D 2 , an integration formula for numerical gapped invariants, like integration of the Berry curvature for the first Chern number, is still missing. At this stage, we simply note the following understanding of numerical gapped invariants for two-dimensional class AIII systems with a corner and three-dimensional class A systems with a hinge. Example 6.3. For a two-dimensional class AIII bulk-edge gapped Hamiltonian on the lattice Z ≥0 × Z ≥0 , our extension of the (off-diagonal part of the) bulk Hamiltonian defines an element I AIII Gapped (H) = [h E ] ∈ K −1 (S 3 ). By Corollary 4.10, the three-dimensional winding number W 3 (h E ) of h E is the same as the Fredholm index index T 0,∞ h = Tr(Π| Ker H 0,∞ ) = I AIII Gapless (H) and accounts for topological corner states. The two-variable rational matrix function f in Example 4.9 provides an example in this class and corresponds to Benalcazar-Bernevig-Hughes' twodimensional model of a second-order topological insulator [9] as discussed in [21].
Example 6.4. For a three-dimensional class A bulk-edge gapped Hamiltonian on the lattice Z ≥0 ×Z ≥0 ×Z, our gapped topological invariant is I A Gapped (H) = [H E ⊕1 N ] ∈ K 0 (S 3 × T). Since H E is a continuous family of self-adjoint invertible matrices, we define a complex vector bundle E onS 3 × T whose fiber at (z, w, t) ∈S 3 × T is Ker(H E (z, w, t) − µ). This vector bundle E is an extension of the Bloch bundle since H E is an extension of the bulk Hamiltonian H. By Theorem 4.1 and the results in [20], the minus of the pairing of second Chern class of this extended Bloch bundle E with fundamental class ofS 3 × T is the same as the spectral flow of the family of self-adjoint Fredholm operators {H 0,∞ (t)} t∈T , therefore accounts for the number of topological hinge states 6 .
Summarizing, we consider a gapped translation invariant single-particle Hamiltonian of finite hopping range on the lattice Z n in each of the ten Altland-Zirnbauer classes. We use Gohberg-Kreȋn theory to factorize the bulk Hamiltonian on the Brillouin torus about two variables z 1 and z 2 . For this purpose, there is a relevant algorithm since our bulk Hamiltonian corresponds to a multivariable rational matrix function on the torus T n [18,12,17]. If all of the partial indices of right matrix factorizations are trivial (equivalently, if compressions of our bulk Hamiltonians onto two half-spaces Z × Z ≥0 × Z n−2 and Z ≥0 × Z × Z n−2 are invertible), we define two topological invariants: One is defined through the restriction of the bulk Hamiltonian onto the lattice (Z ≥0 ) 2 × Z n−2 assuming the Dirichlet boundary condition which provides a (family of) Fredholm operator(s) and its K-class I ♠ Gapless (H) account for topological corner/hinge states. The other is defined as the K-class I ♠ Gapped (H) of the extension of the bulk Hamiltonian ontoS 3 × T n−2 obtained through matrix factorizations, which is our gapped topological invariant (Definition 6.1). There is a relation between these two topological invariants (Theorem 6.2), therefore, corresponding to the gapped topological invariant I ♠ Gapped (H), corner states appear.