Almost commuting unitary matrices related to time reversal
aa r X i v : . [ m a t h . OA ] F e b ALMOST COMMUTING UNITARY MATRICESRELATED TO TIME REVERSAL
TERRY A. LORING AND ADAM P. W. SØRENSEN
Abstract.
The behavior of fermionic systems depends on the ge-ometry of the system and the symmetry class of the Hamiltonianand observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects themathematical behavior of almost commuting Hermitian matricesto depend on two factors. One factor will be the approximate poly-nomial relations satisfied by the matrices. The other factor is whatalgebra the matrices are in, either M n ( A ) for A = R , A = C or A = H , the algebra of quaternions.There are potential obstructions keeping k -tuples of almost com-muting operators from being close to a commuting k -tuple. Weconsider two-dimensional geometries and so this obstruction livesin KO − ( A ). This obstruction corresponds to either the Chernnumber or spin Chern number in physics. We show that if this ob-struction is the trivial element in K -theory then the approximationby commuting matrices is possible. Introduction
Approximate representations.
Consider the two sets of rela-tions S δ X ∗ r = X r k X r X s − X s X r k ≤ δ k X + X + X − I k ≤ δ T ′ δ X ∗ r = X r k X r X s − X s X r k ≤ δ k X + X − I k ≤ δ k X + X − I k ≤ δ that we call the soft sphere relations, S δ , in matrix unknowns X , X , X , and the soft torus relations, T ′ δ , in matrix unknowns X , X , X , X . Our main results concern the operator-norm distance froma representation of S δ or T ′ δ to representations of S or T ′ . We showthat the distance one must move away from the X r to find Hermitianmatrices that are commuting goes to zero as the commutator normgoes to zero.We show that when the X r are taken as variables in M n ( R ), thenrepresentations of S δ are, in a uniform fashion, close to representations of S . When the X r are taken as variables in M n ( C ), there is an ob-struction in Z that dictates the possibility of such an approximation.When the X r are taken as variables in M n ( H ), (the algebra of quater-nions) there is again an obstruction, but now in Z . For X r ∈ M n ( A ) inthese three cases, the obstruction is formally defined to be in KO − ( A )and we prove this is the only obstruction to the desired approximationby commuting Hermitian matrices.The complex case of our results were proven in [17, Corollary 13]and [4, Corollary 6.15], using the techniques of semiprojectivity and K -theory for C ∗ -algebras. The connection of these matrix results tocondensed matter physics was not noticed until many years later [11,9, 10, 19]. Of course the relevance of the K -theory of C ∗ -algebras tocondensed matter physics was known earlier [1].1.2. Structured complex matrices.
Quaternionic matrices arrive indisguise in phsyics via the isometric emdedding χ : M N ( H ) → M N ( C )defined as χ (cid:16) A + B ˆ j (cid:17) = (cid:20) A B − B A (cid:21) for complex matrices A and B . The image of χ can be described as thematrices that commute with the antiunitary operator T ξ = − Zξ where ξ is in C N and Z = Z N = (cid:20) I − I (cid:21) . In a finite model, T is typically playing the role of time-reversal.From a purely mathematical standpoint, it is generally easier to thinkin terms of the dual operation (1.1) (cid:20) A BC D (cid:21) ♯ = (cid:20) D T − B T − C T A T (cid:21) alternatively defined as(1.2) X ♯ = − ZX T Z. The image of χ is the set of matrices with X ∗ = X ♯ . We are usingmathematical notation, so X ∗ refers to the conjugate-transpose of X .Similarly, we think of a real matrix X as a complex matrix for which X ∗ = X T .A good survey paper regarding the equivalence of matrices of quater-nions and structured complex matrices is [5]. This does not addressnorms on M N ( H ). lmost commuting unitary matrices related to time reversal 3 The norm we consider here is that induced by the operator normon M N ( C ). Thus we simply use k A k to denote the operator norm(a.k.a. the spectral norm) of a complex matrix and define for X in M N ( H ) the norm to be k X k = k χ ( X ) k . This norm can be seen to be the norm induced by the action of X on H N , but that is not important. In the same way, we could think aboutthe norm of a real matrix induced by its action on R n but prefer toconsider this norm as being defined via its action on C n .All theorems will be stated in terms of complex matrices, possiblyrespecting an additional symmetry such as being self-dual. This allowsus the freedom to combine two Hermitian matrices X and X into onematrix U = X + iX . If X and X are self-dual, then they are in χ ( M N ( H )), whereas U isself-dual but most likely U ∗ = U ♯ . That is, U cannot be assumed to bein χ ( M N ( H )). We abandon T ′ δ in favor of the relations T δ T δ U ∗ r U r = U r U ∗ r = I k U U − U U k ≤ δ which we no longer apply elements of M N ( H ). (They can be appliedthere, but doing so leads to different question that have less obviousconnections to physics.) Instead we apply them to complex matrices U and U and then add as appropriate U ♯r = U r or U T r = U r . Beyond this point, matrices are assumed to be complex.
Real C ∗ -algebras. Most of our theorems are statements aboutReal C ∗ -algebras. More specifically, we consider C ∗ ,τ -algebras. This isan ordinary (so complex) C ∗ -algebra A with the additional structure ofan anti-multiplicative , C -linear map τ : A → A for which τ ( τ ( a )) = a and τ ( a ∗ ) = τ ( a ) ∗ .We prefer the notation a a τ to keep close to our essential examples( M n ( C ) , T) and ( M n ( C ) , ♯ ). For background of this perspective see[10]. For a reference that uses more traditional notation, see [13]. Mostimportantly, our proofs rely on many results from our previous paper[20]. This deals with the same approximation-by-commuting question,but with the equations D δ D δ X ∗ r = X r k X X − X X k ≤ δ k X k , k X k ≤ lmost commuting unitary matrices related to time reversal 4 in two matrix variables. The underlying geometry is the disk and sothe potential for an obstruction in K -theory is eliminated. Hermitianalmost commuting matrices are always close to commuting Hermitianmatrices. For complex matrices, this is Lin’s theorem [14], and theresult stays true in the real or self-dual case.Our approach is to move from the disk to the sphere along the linesof [17], this involves reformulating the problem as a lifting problem.Then we move from the sphere to the torus using the interplay betweenpush-out diagrams and extensions, generalizing results from [4].1.4. Band-projected position matrices.
Consider a lattice modelwith Hamiltonian H = H ∗ , for a single particle, tight-binding model ona surface in d -space determined by some equations p ( x , . . . , x d ) = 0.The position matrices in the model will be diagonal, so matrices ˆ X r with [ ˆ X r , ˆ X s ] = 0 and p ( ˆ X , . . . , ˆ X d ) = 0. Under some assumptions, es-sentially that we have a spectral gap and local interactions, the Hamil-tonian will approximately commute with the position matrices. Let P denote the projection onto the states below the Fermi energy. Whenwe form the band-projected position matrices X r = P ˆ X r P we arrive atmatrices that are only almost commuting. Depending on the universal-ity class [21] of the system, both H and ˆ X r will have extra symmetries,as will the X r . For example, time reversal invariance will result in thesematrices being self-dual.There are important details (see [10]) needed to correct for the factthat the X r do not have full rank and that the equations need to changeto allow for larger physical size of the lattice when the number of lat-tice size increases. In many interesting cases the result is matrices X ( n )1 , . . . , X ( n ) d of increasing size with (cid:13)(cid:13)(cid:2) X ( n ) r , X ( n ) s (cid:3)(cid:13)(cid:13) → k p ( X , . . . , X d ) k → . When the lattice geometry is the two-torus, the resulting obstructionto the matrix approximation problem is an integer that correspondsto the Chern number. When time reversal invariance is assumed, theextra symmetry leads to an obstruction in Z . This obstruction, forthe self-dual matrix approximation problem, corresponds to the spinChern number used to detect two-dimensional topological insulators.Changing the geometry to a three-torus leads to an obstruction in KO − ( H ) ∼ = Z . There is numerical evidence [10], and the K -homologyarguments of [12] and [6, § III], that this obstruction will be useful fordetecting three-dimensional topological insulators. lmost commuting unitary matrices related to time reversal 5
The obstructions.
When X is any invertible matrix, we definepolar( X ) as the unitary in the polar decomposition. That is, it is the polar part of X or, in terms of the functional calculus,polar( X ) = X ( X ∗ X ) − . Consider H , H and H that are a representation of S δ with δ < .We define B ( H , H , H ) = (cid:20) H H + iH H − iH − H (cid:21) , which will be Hermitian and invertible [16, Lemma 3.2]. The obstruc-tion to being close to commuting is defined as the K -theory classBott( H , H , H ) = (cid:2) polar ( B ( H , H , H )) (cid:3) ∈ K ( C ) . Here we adopt the convention that K ( A ) is defined via homotopyclasses of self-adjoint unitary elements in M n ( A ). This is not the stan-dard view in terms of projections, but is equivalent by a simple shiftand rescaling. This is for A a unital C ∗ -algebra.A more computable description is to use the signature, meaning halfof the number of positive eigenvalues of B minus half the number ofnegative eigenvalues. We call Bott( H , H , H ) the Bott index .(We are off by a minus sign from the definition in [9]. The work in[16] was done without noticing the role of the Pauli spin matrices.)Given a C ∗ ,τ -algebra ( A, τ ) we regard K ( A, τ ) as defined via classesof invertible in M n ( A ) with x ∗ = x and x τ = − x . In [10] we workedwith inveribles with x ∗ = − x and x τ = − x . These are equivalentand the conversion from one picture to the other is done simply bymultiplying by i .When H r = H ♯r for all r , or H r = H T r , the Bott index vanishes so itis possible to approximate by commuting Hermitian matrices. Noticethese nearby commuting matrices will be only approximately self-dualor symmetric. In the self-dual case a new obstruction arises when wetry to approximate by matrices that are at once commuting, Hermitianand self-dual. Here the larger matrix B ( H , H , H ) satisfies( B ( H , H , H )) ♯ ⊗ ♯ = B ( H , H , H ) . Noticing that K ( M N ( C ) , ♯ ⊗ ♯ ) ∼ = Z , the first named author and Hastings defined in [10] the Pfaffian-Bottindex, denoted Pf − Bott( H , H , H ) , lmost commuting unitary matrices related to time reversal 6 as the K -theory class (cid:2) polar ( B ( H , H , H )) (cid:3) ∈ K ( M N ( C ) , ♯ ⊗ ♯ ) ∼ = Z . Much of the work in [19] was in demonstrating that the Pfaffian-Bottindex can be efficiently computed numerically using a Pfaffian. ThePfaffian cannot be applied directly, but conjugating B ( H , H , H ) bya fixed unitary leads to a purely imaginary, skew-symmetric matrix.The sign of the Pfaffian of that matrix tells us which K -class containspolar( B ( H , H , H )).1.6. Main theorems.
We state now two of our four main theorems,along with the complex version. For consistency with [19], we regardthe Bott index as an element of Z and the Pfaffian-Bott index as anelement of the multiplicative group {± } . Theorem 1.1. ([17, Corollary 13])
For every ǫ > there exists a δ > so that whenever matrices H , H , H form a representation of S δ , and Bott( H , H , H ) = 0 , there are matrices K , K , K that form a representation of S and sothat k K r − H r k ≤ ǫ ( r = 1 , , . Theorem 1.2.
For every ǫ > there exists a δ > so that whenever H , H , H are complex symmetric matrices that form a representationof S δ , there are complex symmetric matrices K , K , K that form arepresentation of S and so that k K r − H r k ≤ ǫ ( r = 1 , , . Theorem 1.3.
For every ǫ > there exists a δ > so that whenever H , H , H are self-dual matrices that form a representation of S δ , and Pf − Bott( H , H , H ) = 1 , there are self-dual matrices K , K , K that form a representation of S and so that k K r − H r k ≤ ǫ ( r = 1 , , . Theorems 1.2 and 1.3 settle Conjectures 3 and 4 from [9, § VI.C],while Conjectures 1 and 2 from that paper were settled in our earlierpaper [20].To get theorems about unitary matrices, we utilize an old trick from[16] to turn a representation of T δ into a representation of S δ , butmodified, as in [10], to account for the additional symmetry. We define lmost commuting unitary matrices related to time reversal 7 nonnegative real-valued functions on the circle f, g and h so that f + g + h = 1 and gh = 0 and so that( z, w ) (cid:0) f ( w ) , g ( w ) + { h ( w ) , z } (cid:1) , where {− , −} denotes the anticommutator, is a degree-one mapping ofthe two torus in C to the unit sphere in R × C . The exact choice onlyeffects the relation of δ to ǫ in the theorems below.Given U and U that form a representation of T δ we define H = f ( U ) ,H = g ( U ) + { h ( U ) , U ∗ } + { h ( U ) , U } , and, H = i { h ( U ) , U ∗ } − i { h ( U ) , U } , which then is a representation for S η where η can be taken small when δ is small. The anticommutator { – , – } is used to ensure that U r = U T r or U r = U ♯r propogates to the same symmetry in the H r . Now we defineBott( U , U ) = Bott( H , H , H ) , and Pf − Bott( U , U ) = Pf − Bott( H , H , H ) . It is possible to define these invariants when the U r are only ap-proximately unitary. A simple approach is to compute the invariant asdefined above but using polar( U r ) in place of U r .It is not hard to see that when these indices are nontrivial the approx-imation by commuting matrices of the required form is not possible.See [10]. What is not so apparent is that these indices can be non-trivial. In the case of almost commuting matrices that are unitary,and with no other restrictions, we have the example first considered byVoiculescu [29], with A n the cylic shift on the basis of C n and B n adiagonal unitary. Specifically, when A n = , B n = e πi/n e πi/n . . . e − πi/n , we have Bott( A n , B n ) = 1 , while k [ A n , B n ] k →
0. For a self-dual example, we pair this examplewith its transpose. Using Theorem 2.8 of [10] and the fact the transpose lmost commuting unitary matrices related to time reversal 8 commutes with the functional calculus, one can showPf − Bott (cid:18)(cid:20) A n A T n (cid:21) , (cid:20) B n B T n (cid:21)(cid:19) = − , while the Bott index here is trivial. This is an example of self-dual,almost commuting unitaries that are close to commuting unitaries, butfar from commuting self-dual unitaries.Here then are our other two main theorems, along with the complexversion. Theorem 1.4. ([4, Corollary 6.15])
For every ǫ > there exists a δ > so that whenever matrices U , U form a representation of T δ ,and Bott( U , U ) = 0 , there are matrices V , V that form a representation of T and so that k U r − V r k ≤ ǫ ( r = 1 , . Theorem 1.5.
For every ǫ > there exists a δ > so that whenever U , U are complex symmetric matrices that form a representation ofof T δ , there are complex symmetric matrices V , V that form a repre-sentation of T and so that k U r − V r k ≤ ǫ ( r = 1 , . Theorem 1.6.
For every ǫ > there exists a δ > so that whenever U , U are self-dual matrix representations of T δ , and Pf − Bott( U , U ) = 1 , there are self-dual matrices V , V that are representation of T and sothat k U r − V r k ≤ ǫ ( r = 1 , . Block symmetries in Unstable K -theory Recall Bott( H , H , H ) lives in a matrix algebra twice as big as the H r , and observe that it can be written asBott( H , H , H ) = X H r ⊗ σ r , where σ = (cid:20) (cid:21) , σ = (cid:20) i − i (cid:21) , and σ = (cid:20) − (cid:21) . The σ i , which up to sign and scaling are the Pauli spin matrices, areanti-self-dual. When the H r are complex symmetric, Bott( H , H , H )is anti-self-dual. When the H r are self-dual, Bott( H , H , H ) can be lmost commuting unitary matrices related to time reversal 9 conjugated to complex anti-symmetric. We need to be working in theoriginal formation and so deal with the operation ♯ ⊗ ♯ .Recall that there are, up to isomorphism, just two τ -structures thatcan be put on M n ( C ), the transpose or, when n is even, the dual, see[13, § M n ( C ) ⊗ M ( C ) can force symmetries on one of its blocks. This isLemma 2.3. Lemma 2.1.
Suppose ( M n ( C ) , τ ) is a C ∗ ,τ -algebra, and consider thelarger C ∗ ,τ -algebra ( M n ( C ) , τ ⊗ ♯ ) , meaning that on M n ( C ) we are using the τ -operation (cid:20) A BC D (cid:21) τ ⊗ ♯ = (cid:20) D τ − B τ − C τ A τ (cid:21) . Within the group (cid:8) W ∈ M n ( C ) (cid:12)(cid:12) det( W ) = 1 , W ∗ = W − = W τ ⊗ ♯ (cid:9) , the set of W = (cid:20) A B − B ∗ τ A τ ∗ (cid:21) , for which both A and B are invertible, is a dense open subset.Proof. Up to isomorphism of C ∗ ,τ -algebras, there are two cases, τ = Tand τ = ♯ . In both cases, the openness is clear.Suppose τ = T, which means τ ⊗ ♯ = ♯ . First we not that theinvertiblity of A and B holds in one case, specifically(2.1) W = 1 √ (cid:20) I I − I I (cid:21) , which is a symplectic unitary, i.e. W ♯ = W ∗ = W − .All symplectic matrices have determinant 1, so we can ignore thedeterminant condition. We know from Lie theory that every symplec-tic unitary is e H for a matrix H with H ∗ = H ♯ = − H . Given anysymplectic unitary W = (cid:20) A B − B A (cid:21) , we can find a matrix H with H ∗ = H ♯ = − H such that W = e H W .Thus we have an analytic path W t = e tH W from W to W , with W t a symplectic unitary at every t . Let the upper blocks of W t be A t and B t . Considering the power series for e tH we find convergent powerseries for the scalar paths det( A t ) and det( B t ). As these analytic paths lmost commuting unitary matrices related to time reversal 10 are nonzero around t = 1, neither can vanish on any open interval. Wecan choose t close to 0 to find W t with A t and B t both invertible.We prove the τ = ♯ case similarly, starting with the same example, W as in (2.1), but now with n = 2 N . A real orthogonal matrix ofdeterminant one is e H for a matrix with H ∗ = H T = − H . Translatingthis fact via the isomorphism Φ from ( M N ( C ) , ♯ ⊗ ♯ ) to ( M N ( C ) , T),which is just conjugation by a unitary (see [10, Lemma 1.3] or equiv-alently (3.2)), we see that when W is a unitary with determinant oneand W ∗ = W ♯ ⊗ ♯ , there is a matrix H with H ∗ = H ♯ ⊗ ♯ = − H so that e H = W . Therefore if we are given W a unitary with W ∗ = W ♯ ⊗ ♯ we can find H with H ∗ = H ♯ ⊗ ♯ = − H such that W = e H W , and sowe have an analytic path W t = e tH W of unitaries, and one can checkdirectly, or using Φ, that W ∗ t = W ♯ ⊗ ♯t . The rest of the argument isexactly as in the previous case. (cid:3) Lemma 2.2. If a and b are invertible elements of a unital C ∗ -algebraand aa ∗ + bb ∗ = 1 then polar( a ∗ b ) = (polar( a )) ∗ polar( b ) . Proof.
We use the formula polar( x ) = x ( x ∗ x ) − and compute a ∗ b ( b ∗ aa ∗ b ) − = a ∗ b ( b ∗ (1 − bb ∗ ) b ) − = a ∗ (1 − bb ∗ ) − ( bb ∗ ) − b = ( a ∗ a ) − a ∗ b ( b ∗ b ) − . (cid:3) Lemma 2.3. If W = (cid:20) A B − B ∗ τ A τ ∗ (cid:21) , is a unitary in M n ( C ) with W ∗ = W τ ⊗ ♯ and both A and B are invert-ible, then (polar( A )) ∗ polar( B ) is fixed by τ .Proof. From the fact that W is unitary we deduce AA ∗ + BB ∗ = I and A ∗ B = B τ A τ ∗ , so ( A ∗ B ) τ = A ∗ B . The last lemma tell us(polar( A )) ∗ polar( B ) = polar( A ∗ B ) . From the definition of the polar part in terms of functional calculus wefind (polar( x )) τ = polar( x τ ) and therefore(polar( A ∗ B )) τ = polar(( A ∗ B ) τ ) = polar( A ∗ B ) . (cid:3) lmost commuting unitary matrices related to time reversal 11 Structure Diagonalization and KO Hermitian anti-self-dual invertibles.
The K -theory we willneed to track turns out to be K . In the case of symmetric matri-ces, we will need K of M n ( C ) ⊗ M ( C ) ∼ = M n ( C ). The fact that K ( M N ( C ) , ♯ ) = 0 can be given the concrete realization that for anytwo Hermitian anti-self-dual matrices that are approximately unitary,there is a symplectic unitary that approximately conjugates one to theother. Lemma 3.1.
Suppose X in M N ( C ) is Hermitian and anti-self-dual.Then there exists a symplectic unitary W and a diagonal matrix D withnonnegative real diagonal entries so that X = W (cid:20) D − D (cid:21) W ∗ . Proof.
The matrix Y = − iX has Y ♯ = Y ∗ so we may apply re-sults about matrices of quaternions. Since Y is normal, indeed skew-Hermitian, the spectral theorem for matrices of quaternions [5, Theo-rem 3.3] gives us a symplectic unitary W so that X = W (cid:20) D − D (cid:21) W ∗ . This follows more easily form a different version of the quaternionicspectral theorem given in [18, Theorem 2.4(2)], see also [5, Page 87].We need only adjust W and D so that the diagonal of D ends up non-negative. We can swap the j -th diagonal element of D with the j -thdiagonal element of − D as needed by utilizing the formula (cid:20) ii (cid:21) (cid:20) λ − λ (cid:21) (cid:20) ii (cid:21) ∗ = (cid:20) − λ λ (cid:21) . (cid:3) Theorem 3.2.
Suppose k S − I k < and S ∗ = S and S ♯ = − S . Thenthere is a symplectic unitary W so that (cid:13)(cid:13)(cid:13)(cid:13) S − W (cid:20) I − I (cid:21) W ∗ (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) S − I (cid:13)(cid:13) . Proof.
Lemma 3.1 tells us S = W ∗ (cid:20) D − D (cid:21) W, lmost commuting unitary matrices related to time reversal 12 for some symplectic unitary W and some diagonal matrix D with non-negative real entries. Let δ = k S − I k . We compute (cid:13)(cid:13) D − I (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:20) D − D (cid:21) − (cid:20) I I (cid:21)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) S − I (cid:13)(cid:13) = δ, so the diagonal elements of D are in (cid:8) λ (cid:12)(cid:12) − δ ≤ λ ≤ δ (cid:9) ⊆ (cid:8) λ (cid:12)(cid:12) − δ ≤ λ ≤ δ (cid:9) . The result now follows. (cid:3)
Coupling two dual operations.
In the self-dual case, the K -theory is calculated in M N ( C ) ⊗ M ( C ) ∼ = M N ( C ) and the operationthat specifies the symmetry on the Bott matrix is ♯ ⊗ ♯ . In terms of4 n -by-4 n matrices, ♯ ⊗ ♯ is T conjugated by a unitary. There are manyidentifications of M N ( C ) ⊗ M ( C ) with M N ( C ) and the one we useoperates via B ⊗ (cid:20) (cid:21) (cid:20) B (cid:21) . Of course B itself will often be written out in terms of four N -by- N blocks. We are using Z = Z N = (cid:20) I − I (cid:21) ∈ M N ( C ) , and the operation ♯ ⊗ ♯ works out as (cid:20) A BC D (cid:21) ♯ ⊗ ♯ = (cid:20) D ♯ − B ♯ − C ♯ A ♯ (cid:21) . The symmetry X ♯ ⊗ ♯ = X ∗ means(3.1) X = A A B B A A B B − B B A − A B − B − A A . The isomorphism we need, cf. [10, Lemma 1.3], is(3.2) Φ : ( M N ( C ) , ♯ ⊗ ♯ ) → ( M N ( C ) , T)defined by Φ ( X ) = U XU ∗ , where X is 4 N -by-4 N and U = 1 √ I ⊗ I − iZ N ⊗ Z N ) . lmost commuting unitary matrices related to time reversal 13 Any X in the form (3.1) is conjugate to a matrix with real entries, andso det( X ) ∈ R . This is the trick that allows us to get our Z -invariantas the sign of a determinant of some invertible X that is full of complexnumbers.3.3. Hermitian anti-symmetric invertibles.
Because of the cou-pling of the dual operations, and the doubling of the matrix sizes inthe proofs of Theorems 4.1 and 4.2, we will be interested in the K -group of ( M n ( C ) , T). The groups is Z , and this fact can be given thefollowing concrete realization. The hermitian anti-symmetric matricesfall into two classes: those that can be approximately conjugated byan element of SL n ( R ) to S = − n i ( − n ( − i ) 0 0 i − i i − i i − i and those that cannot. Note that Pf( S ) = ( − n · i n = 1. The signof the Pfaffian can decide which class a matrix X is in. Theorem 3.3.
Suppose k S − I k < and S ∗ = S and S T = − S forsome S in M n ( C ) . Then there is a real orthogonal W of determinantone with k S − W S W ∗ k ≤ (cid:13)(cid:13) S − I (cid:13)(cid:13) if and only if Pf( S ) is strictly positive.Proof. We first prove the “if” part. Let X = − iS and observe that X ∗ = X T = − X . It follows from [10, Theorem 9.4] that we can finda real orthogonal matrix U and real numbers a , a , . . . , a n such that X = U DU T , where D = a − a a − a a n − a n . lmost commuting unitary matrices related to time reversal 14 We may assume that the sign of a is ( − n and that a , a , . . . a n > X in two ways. Using that X = − iS we get Pf( X ) = Pf( − iS ) = ( − i ) n Pf( S ) = ( − n Pf( S ) . Using instead that X = U DU T we getPf( X ) = Pf( U DU T ) = det( U )Pf( D ) = det( U ) a a · · · a n . Hence det( U ) must be positive, in particular it is 1. Finally we see that k S − U S U ∗ k = k iD − S k ≤ k ( iD ) − I k = k S − I k . Suppose now there is W in SL n ( R ) so that k S − W S W ∗ k < . Then S = W ∗ SW satisfies S ∗ = S and S T1 = − S and k S − S k < S t = (1 − t ) S + tS is within 1 of S and S is a unitary, soeach S t is invertible and skew-symmetric. Therefore the path of realnumbers Pf( S t ) cannot change sign. Since Pf( S ) = 1 we must havePf( S t ) > t . (cid:3) Theorem 3.4.
Suppose k S − I k < and S ∗ = S and S ♯ ⊗ ♯ = − S forsome S in M n ( C ) . Then there is a unitary W with W ♯ ⊗ ♯ = W ∗ with (cid:13)(cid:13)(cid:13)(cid:13) S − W (cid:20) I − I (cid:21) W ∗ (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) S − I (cid:13)(cid:13) if and only if the K class represented by S is trivial.Proof. We will derive this from Theorem 3.3 via the isomorphism Φfrom (3.2). Recall this satisfies Φ( X ♯ ⊗ ♯ ) = (Φ( X )) T and, being a ∗ -isomorphism, satisfies Φ( X ∗ ) = (Φ( X )) ∗ . A short computation (withblocks in two different sizes) tells usPf (cid:18) Φ (cid:18)(cid:20) I − I (cid:21)(cid:19)(cid:19) = Pf iI − iI iI − iI = 1 . We apply Theorem 3.3 to Φ (cid:18)(cid:20) I − I (cid:21)(cid:19) and find a real orthogonal W so that Φ (cid:18)(cid:20) I − I (cid:21)(cid:19) = W S W ∗ . The conditions on S translates to k Φ( S ) − I k <
1, Φ( S ) ∗ = Φ( S ),and Φ( S ) T = − Φ( S ). If the K -class of S is trivial then Pf (Φ( S )) is lmost commuting unitary matrices related to time reversal 15 positive and so there is a real orthogonal W with k Φ ( S ) − W S W ∗ k ≤ (cid:13)(cid:13) Φ( S ) − I (cid:13)(cid:13) = (cid:13)(cid:13) S − I (cid:13)(cid:13) . The desired unitary is then W = Φ − ( W W ∗ ). If on the other handthere is a unitary W ∈ M n ( C ) as in the statement of the theorem, thenwe see that Φ( W ) W almost conjugates Φ( S ) to S . Hence Pf (Φ( S ))is positive, and so the K -class of S is trivial. (cid:3) From K − to K . Behind our definition of the Bott index andthe Pfaffian-Bott index is a specific generator of K − of ( C ( S ) , id).The associated R ∗ -algebra here is C ( S , R ). We skip over the actualdefition of K − and utilize a very important part of Bott periodicitywhich is that there is a natural isomorphism [24] K − ( A ) ∼ = K ( A ⊗ H ) , for R ∗ -algebras, and which in terms of C ∗ ,τ -algebras is K − ( A, τ ) ∼ = K ( A ⊗ M ( C ) , τ ⊗ ♯ ) . We take then K ( A ⊗ M ( C ) , τ ⊗ ♯ ) as the definition of K − ( A, τ ).With this convention, the generator of K − ( C ( S ) , id) is the class ofskew- τ invertible Hermitian elements represented by b = (cid:20) x y + izy − iz − x (cid:21) where x , y and z are the coodinate functions if we regard S as theunit sphere in R . A basic fact in K -theory is that K − (cid:0) C ( S ) , id (cid:1) ∼ = Z . If we have an actual ∗ - τ -homomorphism ϕ from ( C ( S ) , id) to some( A, τ ), as in the proofs that follow, then calculating K − ( ϕ ) is just amatter of following b over to where it lands in ( M ( A ) , τ ⊗ ♯ ), meaning (cid:20) ϕ ( x ) ϕ ( y ) + iϕ ( z ) ϕ ( y ) − iϕ ( z ) − ϕ ( x ) (cid:21) . Spherical to cylindrical coordinates
We consider various inclusions of commutative C ∗ ,τ -algebras, all in-duced by surjections. The first we will encounter is(4.1) ι : C ( S , id) ֒ → C ( S × [ − , , id)which is unital and induced by (cid:0) e πiθ , t (cid:1) (cid:16) √ − t cos( θ ) , √ − t sin( θ ) , t (cid:17) . lmost commuting unitary matrices related to time reversal 16 We consider C ( S × [ − , , id) as universal for a unitary v and a positivecontraction k that commute. We also need relations for the τ operationto create the identity involution on the cylinder, so we require v τ = v and k τ = k . In term of generators and relations, the inclusion (4.1)operates via h (cid:0) − k (cid:1) ( v ∗ + v ) ,h (cid:0) − k (cid:1) ( iv ∗ − iv ) ,h k. Generators and relations for real C ∗ -algebras have only been consid-ered implicitly, as in [7, § II]. The relations we need are rather basic, in-volving only real C ∗ -algebras that are commutive or finite-dimensional,so we will not explore this topic formally here. It has been investigatedin [26]. Theorem 4.1.
Suppose ( d n ) is a sequence of natural numbers and that ϕ : C ( S , id) → Y ( M d n ( C ) , T) .M ( M d n ( C ) , T) , is a unital ∗ - τ -homomorphism. Then there exists a unital ∗ - τ -homo-morphism ψ so that C ( S × [ − , , id) ψ C ( S , id) ι ϕ Q ( M d n ( C ) , T) / L ( M d n ( C ) , T) , commutes.Proof. Let π : Q ( M d n ( C ) , T) ։ Q ( M d n ( C ) , T) / L ( M d n ( C ) , T) , bethe quotient map, and let H r be the image under ϕ of the generators h r of C ( S ). Select lifts under π of the H r , so matrices H n,r ∈ M d n ( C ).Taking averages and fiddling with functional calculus, we can assumethat these are contractions with(4.2) H n,r = H ∗ n,r = H T n,r . Let π denote the quotient map. Since H r = π ( h H ,r , H ,r , . . . i )we have that as n → ∞ (4.3) k H n,r H n,s − H n,s H n,r k → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r H n,r − I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → lmost commuting unitary matrices related to time reversal 17 When defining ψ by where to send generators we can ignore any initialsegment. Therefore we may assume, without loss of generality, that(4.4) k H n,r H n,s − H n,s H n,r k <
14 and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X r H n,r − I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < , for all n . We apply [9, Lemma 3.2] to S n = (cid:20) H n, H n, + iH n, H n, − iH n, − H n, (cid:21) to find k S n − I k < n , and k S n − I k →
0. By (4.2) we have S ∗ n = S n and S ♯n = − S n .Let δ n be some numbers with δ n → (cid:13)(cid:13) S n − I (cid:13)(cid:13) < δ n < . Theorem 3.2 provides us with A n and B n so that W n = (cid:20) A n B n − B n A n (cid:21) is a symplectic unitary (so of determinant one) with (cid:13)(cid:13)(cid:13)(cid:13) S n − W ∗ n (cid:20) I − I (cid:21) W n (cid:13)(cid:13)(cid:13)(cid:13) < δ n . Lemma 2.1 allows us to perturb A n and B n a little so as to keep theseconditions and have A n and B n invertible. Lemmas 2.2 and 2.3 tells uspolar( A ∗ n B n ) = (polar( A n )) ∗ polar( B n ) , and (polar( A ∗ n B n )) T = polar( A ∗ n B n ) . Now we follow the procedure in [9, Section III.C], which has a historygoing back to [15, Section 2]. However, we work in terms of A = π ( h A , A , . . . i ) ,B = π ( h B , B , . . . i ) ,S = (cid:20) H H + iH H − iH − H (cid:21) , and W = (cid:20) A B − B ∗ τ A ∗ τ (cid:21) , lmost commuting unitary matrices related to time reversal 18 as we are not aiming for a quantitative result. We use τ to denote theoperation in the quotient induced by all the matrix transpose opera-tions. We can see that W is a unitary with(4.5) S = W ∗ (cid:20) − (cid:21) W. From
W W ∗ = I we derive AA ∗ + BB ∗ = 1 and AB τ = BA τ . From(4.5) we derive (cid:20) A ∗ B ∗ (cid:21) (cid:20) A B (cid:21) = (cid:20) + H H + i H H − i H − H (cid:21) , so we find A ∗ A = 12 + 12 H ,A ∗ B = 12 H + i H , and B ∗ B = 12 − H . Combining two of these we find A ∗ A + B ∗ B = I. Let Z = π ( h polar( A ) , polar( A ) , . . . i ) , and V = π ( h polar( B ) , polar( B ) , . . . i ) , so that A = Z ( A ∗ A ) = ( AA ∗ ) Z, and B = V ( B ∗ B ) = ( BB ∗ ) V. Then V (cid:18)
12 + 12 H (cid:19) V ∗ = V A ∗ AV ∗ = 1 − V B ∗ BV ∗ = 1 − BB ∗ = AA ∗ and Z ∗ AA ∗ Z = A ∗ A = 12 + 12 H . Put together, these tell us V ∗ Z (cid:18)
12 + 12 H (cid:19) Z ∗ V = 12 + 12 H , and so the unitary U = Z ∗ V commutes with H . Since U = π ( h (polar( A )) ∗ polar( B ) , (polar( A )) ∗ polar( B ) , . . . i )we have the additional symmetry U τ = U. Therefore we can define ψ : C ( S × [ − , , id) → Y ( M d n ( C ) , T) .M ( M d n ( C ) , T) , lmost commuting unitary matrices related to time reversal 19 by sending the universal generators of C ( S × [ − , , id) to U and H .This makes the diagram commute because U (cid:0) − H (cid:1) = 2 Z ∗ V ( B ∗ B ) ( A ∗ A ) = 2 Z ∗ (1 − BB ∗ ) B = 2 Z ∗ ( AA ∗ ) B = 2 A ∗ B = H + iH . (cid:3) Theorem 4.2.
Suppose d n is a sequence of natural numbers and that ϕ : C ( S , id) → Y ( M d n ( C ) , ♯ ) .M ( M d n ( C ) , ♯ ) is a unital ∗ - τ -homomorphism. If K − ( ϕ ) = 0 then there exists a unital ∗ - τ -homomorphism ψ so that C ( S × [ − , , id) ψ C ( S , id) ι ϕ Q ( M d n ( C ) , ♯ ) / L ( M d n ( C ) , ♯ ) commutes.Proof. The proof begins as before, with H n,r in M d n ( C ) and with (4.2)replaced by H n,r = H ∗ n,r = H ♯n,r . After making the reduction to get (4.4), the Pfaffian-Bott index is welldefined for all n . The assumption on the K -theory of ϕ tells us thatthis index is zero for large n . Further truncating the sequences, we canreduce to the case where S n represents the trivial K class for all n .Let δ n be some numbers with δ n → (cid:13)(cid:13) S − I (cid:13)(cid:13) < δ n < . Theorem 3.4 provides us with W n = (cid:20) A n B n − B n A n (cid:21) that is a unitary of determinant one with W ♯ ⊗ ♯n = W ∗ n and (cid:13)(cid:13)(cid:13)(cid:13) S n − W ∗ n (cid:20) I − I (cid:21) W n (cid:13)(cid:13)(cid:13)(cid:13) < δ n . lmost commuting unitary matrices related to time reversal 20 Figure 4.1.
The map of Ω [1] onto Ω.Lemma 2.1 again allows us to reduce to the case where A n and B n areinvertible. Lemmas 2.2 and 2.3 tell uspolar( A ∗ n B n ) = (polar( A n )) ∗ polar( B n ) , and (polar( A ∗ n B n )) ♯ = polar( A ∗ n B n ) . The rest of the proof goes though unchanged, although τ in the quotientis that derived from the sequence of ♯ operations. (cid:3) We now wish to proceed as in [4, Section 6.3], and so need a way to“poke holes” in open patches. Let Ω denote the open unit diskΩ = (cid:8) ( x, y ) (cid:12)(cid:12) x + y < (cid:9) and Ω [1] = (cid:8) ( x, y ) (cid:12)(cid:12) ≤ x + y < (cid:9) . We have a proper surjective continuous map Ω [1] → Ω srinking theinner circle to a point. We think of Ω [1] as Ω with replaced by acircle. See Figure 4.1. This gives us a map ι : C (Ω) ֒ → C (cid:0) Ω [1] (cid:1) . Corollary 4.3.
Suppose ϕ : C (Ω , id) → Y ( M d n ( C ) , τ n ) .M ( M d n ( C ) , τ n ) lmost commuting unitary matrices related to time reversal 21 is a ∗ - τ -homomorphism with τ n = T for all n or τ n = ♯ and d n evenfor all n . If K − ( ϕ ) = 0 then there exists a ∗ - τ -homomorphism ψ sothat C (Ω [1] , id) ψ C (Ω , id) ι ϕ Q ( M d n ( C ) , τ n ) / L ( M d n ( C ) , τ n ) commutes.Proof. The inclusion C (Ω , id) ֒ → ( C (Ω , id)) ∼ into the unitization isan isomorphism on K − , (essentially by definition since K − ( R ) = 0).Therefore it suffices to prove the equivalent unital extension problemwhere we unitize the two commutative C ∗ ,τ -algebras: C ( D , id) e ψ C ( S , id) ι e ϕ Q ( M d n ( C ) , τ n ) / L ( M d n ( C ) , τ n )Here ι comes from the degree-one map D → S that sends the originto the north pool and the boundary circle to the south pole. We havethe factorization C ( S ) ι ι C ( D , id) C ( S ⊗ [ − , , id)and so this follows from Theorems 4.1 and 4.2. (cid:3) To prove our results for the torus geometry, we need to know aboutamalgamated products of C ∗ ,τ -algebras. For the sphere geometry, weare ready to give a proof. That is, we now prove Theorems 1.2 and 1.3.Given ϕ : C ( S ) → Y ( M d n ( C ) , τ n ) .M ( M d n ( C ) , τ n )with trivial K we have, as we saw in the proof of Corollary 4.3, anextension to a ∗ - τ -homomorphims from C ( D ). This we can lift to theproduct by the main result in [20]. This solves the lifting problem, atleast when the K -theory allows it, for the map from C ( S ). We leaveto the reader the usual conversion of Theorems 1.2 and 1.3 into a liftingproblem via generators and relations. lmost commuting unitary matrices related to time reversal 22 Amalgamated products of C ∗ ,τ -algebras Define Υ A : A → A op to be the identity map on the underlying set,so a ∗ -anti-homomorphism. We have ( A op ) op = A. Lemma 5.1.
Suppose
C, A and A are C ∗ -algebras, θ : C → A and θ : C → A are ∗ -homomorphisms and consider the associated amal-gamated product A ∗ C A and canonical ∗ -homomorphisms ι j : A j → A ∗ C A . If ϕ j : A → D and ϕ : A → D are ∗ -anti-homomorphismssuch that ϕ ◦ θ = ϕ ◦ θ , then there is a unique ∗ -anti-homomorphism Φ : A ∗ C A → D such that Φ ◦ ι j = ϕ j . Proof.
Both Υ D ◦ ϕ and Υ D ◦ ϕ are ∗ -homomorphisms, and Υ D ◦ ϕ ◦ θ = Υ D ◦ ϕ ◦ θ . There is a unique ∗ -homomorphism Ψ : A ∗ C A → D op such that Ψ ◦ ι j = Υ D ◦ ϕ j . Let Φ = Υ D op ◦ Ψ. This is a ∗ -anti-homomorphism andΦ ◦ ι j = Υ D op ◦ Ψ ◦ ι j = Υ D op ◦ Υ D ◦ ϕ j = ϕ j . If Φ ′ : A ∗ C A → D is also a ∗ -anti-homomorphism with Φ ′ ◦ ι j = ϕ j then Υ D ◦ Φ ′ is a ∗ -homomorphism andΥ D ◦ Φ ′ ◦ ι j = Υ D ◦ ϕ j , therefore Υ D ◦ Φ ′ = Ψ and soΦ ′ = id D ◦ Φ ′ = Υ D op ◦ Υ D ◦ Φ ′ = Υ D op ◦ Ψ = Φ . (cid:3) What the lemma shows is that not only is A ∗ C A universal for ∗ -homomorphisms out of A and A , it is also universal for anti- ∗ -homomorphisms. We will use this fact to put a τ -structure on theamalgamated product. Theorem 5.2.
Suppose ( C, τ ) , ( A , τ ) and ( A , τ ) are C ∗ ,τ -algebrasand θ : C → A and θ : C → A are ∗ - τ -homomorphisms. The C ∗ ,τ -algebra A ∗ C A becomes a C ∗ ,τ -algebra with the unique operation τ making both ι j : A j → A ∗ C A into ∗ - τ -homomorphisms. Moreover, ( A ∗ C A , τ ) is the amalgamated product of ( A , τ ) and ( A , τ ) over ( C, τ ) .Proof. We have anti-homomorphisms T : C → C and T j : A j → A j defined by T ( c ) = c τ and T j ( a ) = a τ j . The statement that θ j is a ∗ - τ -homomorphism translates to θ j ◦ T = T j ◦ θ j . So we have ι ◦ T ◦ θ = ι ◦ θ ◦ T = ι ◦ θ ◦ T = ι ◦ T ◦ θ . Therefore there is a unique ∗ -anti-homomorphism T : A ∗ C A → A ∗ C A such that T ( ι j ( a )) = ι j ( a τ j ). Certainly T ◦ T is a ∗ -homomorphisms, lmost commuting unitary matrices related to time reversal 23 and since it fixes ι j ( a ) it is the identity map. We thus can make A ∗ C A into a C ∗ ,τ -algebra by x τ = T ( x ). For example,( ι ( a ) ι ( b ) ι ( c )) τ = ι ( c τ ) ι ( b τ ) ι ( a τ ) . If we are given ϕ j : A j → B , two ∗ - τ -homomorphisms such that ϕ ◦ θ = ϕ ◦ θ , then there is a unique ∗ -homomorphism Φ : A ∗ C A → B such that Φ( ι j ( a )) = ϕ j ( a ) for all a in A j . To finish, we must showthat Φ is actually a ∗ - τ -homomorphism. Given a product w = ι ( a ) ι ( b ) ι ( a ) ι ( b ) · · · ι ( a n ) ι ( b n )we find Φ( w τ ) = Φ ( ι ( b τ n ) ι ( a τ n ) · · · ι ( b τ ) ι ( a τ ))= ϕ ( b τ n ) ϕ ( a τ n ) · · · ϕ ( b τ ) ϕ ( a τ )= ϕ ( b n ) τ ϕ ( a n ) τ · · · ϕ ( b ) τ ϕ ( a τ ))= ( ϕ ( a ) ϕ ( b ) · · · ϕ ( a n ) ϕ ( b n )) τ = (Φ ( ι ( a ) ι ( b ) · · · ι ( a n ) ι ( b n ))) τ = (Φ ( w )) τ . For the other three types of words (ending in ι ( A ) or beginning in ι ( A )) we also find Φ( w τ ) = Φ( w ) τ . Since Φ is linear and continuous,we conclude Φ( x τ ) = Φ( x ) τ for all x in A ∗ C A . (cid:3) We can talk about push-out diagrams instead of amalgamated prod-ucts. If C is large relative to A or A this tends to be a more fittingdescription. What Theorem 5.2 shows is that when a diagram DA B,C is a diagram of ∗ - τ -homomorphisms and C ∗ ,τ -algebras, if it is a push-out in the category of C ∗ -algebras then it is a push-out in the categoryof C ∗ ,τ -algebras. Definition 5.3.
We say that a ∗ - τ -homomorphism is proper if it isproper as a ∗ -homomorphism.Recall (or see [4]) that a ∗ -homomorphism ϕ : A → B is said to beproper if for some (equivalently every) approximate unit e λ of A theimage ϕ ( e λ ) is an approximate unit for B . Using Cohen factorizationwe find that this is equivalent to the condition B = ϕ ( A ) B , see [22]. lmost commuting unitary matrices related to time reversal 24 Figure 6.1.
How we treate each 2-cell in a 2-dimensional CW complex X . Theorem 5.4.
If a diagram of extensions A C B A α C B of C ∗ ,τ -algebras is given, with α proper, then the left square is a push-out.Proof. This is an immediate corollary of this statement in the categoryof C ∗ -algebras [4, Corollary 4.3] and Theorem 5.2. (cid:3) The sphere, torus and other 2D situations
Suppose X is a two-dimensional, finite CW complex. Assume the2-cells are given as unit squares, and on that unit square a lattice ofsize m , at the center point of every square created by the lattice gridwe insert a circle and so create a space X [ m ] . Let Γ m denote the closedsubset of X corresponding to replacing each 2-cell with the 1-D mesh.Consult figure 6.1. Theorem 6.1.
Suppose X is a finite two-dimensional CW complexand that ϕ : C ( X, id) → Y ( M d n ( C ) , τ n ) .M ( M d n ( C ) , τ n ) is a unital ∗ - τ -homomorphism with τ n = T for all n or τ n = ♯ and d n even for all n . If K − ( ϕ ) = 0 then there exists a ∗ - τ -homomorphism ψ lmost commuting unitary matrices related to time reversal 25 so that C ( X [ n ] , id) ψ C ( X, id) ι ϕ Q ( M d n ( C ) , τ n ) / L ( M d n ( C ) , τ n ) , commutes.Proof. Suppose Y is an open set in X and that Y is homeomorphic tothe open unit disk Ω. Let Z be the space we get from X if we removethe closed set in Y that corresponds to the open disk of radius atthe center of Ω. Denote by W the open disc with a hole in Z , withnotational abuse W = Z ∩ Y . Map Z onto X by a continuous functionthat fixes points in X \ W and inside W it corresponds to the surjection( W =) (cid:8) ( x, y ) (cid:12)(cid:12) ≤ x + y < (cid:9) ։ (cid:8) ( x, y ) (cid:12)(cid:12) x + y < (cid:9) (= Ω) , that shrinks the hole in middle to a point. Let ι : C ( X, id) ֒ → C ( Z, id)denote the induced inclusion.We can complete the proof by induction, if we can show the followingextension property C ( Z, id) ψ C ( X, id) ι ϕ Q ( M d n ( C ) , τ n ) / L ( M d n ( C ) , τ n ) , where ϕ is assumed unital and ψ is required to be unital.By using homeomorphisms we can replace W by Ω [1] and Y by Ω.Therefore we get a commutative diagram of C ∗ ,τ -algebras0 C (Ω [1] , id) C ( Z, id) C ( X \ Y, id) 00 C (Ω , id) α C ( X, id) C ( X \ Y, id) 0 , with exact rows. The map Ω [1] → Ω inducing α is proper, hence α isproper and by Theorem 5.4 the left square is a push-out. By Corol-lary 4.3 we can find a ∗ - τ -homomorphism λ such that C (Ω [1] , id) λ C (Ω , id) α C ( X, id) φ Q ( M d n ( C ) , τ n ) / L ( M d n ( C ) , τ n ) , lmost commuting unitary matrices related to time reversal 26 commutes. By the universal property of push-outs, λ and φ combineto define a ∗ -homomorphism ψ : C ( Z, id) → Y ( M d n ( C ) , τ n ) .M ( M d n ( C ) , τ n ) , that extends φ . As the unit in C ( Y ) comes from the unit in C ( X ) itis automatic that ψ is unital. (cid:3) Theorem 6.2.
Suppose X is a finite two-dimensional CW complex.Then for any unital ∗ - τ -homomorphism ϕ : C ( X, id) → Y ( M d n ( C ) , τ n ) .M ( M d n ( C ) , τ n ) , with τ n = T for all n or τ n = ♯ and d n even for all n . If K − ( φ ) = 0 then there exists a unital ∗ - τ -homomorphism ψ making the followingdiagram commute: Q ( M d n ( C ) , τ n ) π C ( X, id) ϕ ψ Q ( M d n ( C ) , τ n ) / L ( M d n ( C ) , τ n ) . Proof.
Let F be a finite generating set for C ( X ). By an intertwiningargument as in the proof of Theorem 3.1 in [3] we can show it sufficesto find for each ǫ > ψ so that k ψ ( f ) − π ◦ ϕ ( f ) k < ǫ for all f in F .Using Theorem 6.1 we can extend ϕ to C ( X [ m ] , id) for large m and thenby retracting X [ m ] to Γ n we get that ϕ approximately factors through C (Γ n , id). A map that factors through C (Γ n , id) will lift exactly, bythe semiprojectitity of C (Γ n , id) which was established in our previouspaper, [20]. (cid:3) If we can describe C ( X ) by generators and relations, we get corol-laries about approximation of tuples of real or self-dual matrices. Twospaces for which this works well are the familiar sphere and torus.We have thus proven the two remaining main theorems, Theorems1.5 and 1.6.7. Controlling maximum Wannier spread
Commuting sets of Hermitian matrices can be simultaneously diag-onalized by a unitary matrix. Thus our main results have equivalentformulations in terms of simultaneous approximate diagonalization. Si-multateous approximate diaganalization arises in signal processing [2]and the resulting algorithms have been unitilized in first principalsmolecular dymamics [8]. lmost commuting unitary matrices related to time reversal 27
There are many ways to measure the failure of a set of matrices tobe diagonal. Essentially every unitarly invariant norm of the matricesgives a means to measure this. If the matrices can be interpreted asposition operators, one can instead ask for “exponential localization”which really only makes sense if we are considering a class of finitemodels of increasing lattice size. We bring this up because it is proba-bly the most physically relevant measure. Naturally, it is hard to evendefine, much less compute numerically. Within the choices of norms,the operator norm would be the most physically relevant, while some-thing like the Frobenius norm would be the easiest to incorporate intoan efficient algorithm.In condensed matter physics, one often looks for a basis for an energyband that is well localized. When computed numercially, generallyone attempts to minimize the total
Wannier spread, which correspondsto minimizing off-diagonal parts in the Frobenius norm. We considertheoretical bounds on minimizing the maximum
Wannier spread.Given X , . . . , X d Hermitian matrices in M n ( C ) and a unit vector b ,the Wannier spread of b with respect to the set X of these of X r wedefine as σ X ( b ) = d X r =1 (cid:10) X r b , b (cid:11) − h X r b , b i . Given an orthonormal subset B = { b , . . . , b k } of C n we define its totalWannier spread with respect to X as X j σ X ( b j ) , and its maximum Wannier spread with resect to X as µ X ( B ) = max j σ X ( b j ) . For X = { X , . . . , X d } and Y = { Y , . . . , Y d } , both sets of Hermitianmatices on C n , define kX k = max r k X r k , and dist ( X , Y ) = max r k X r − Y r k . Lemma 7.1.
Suppose X = { X , . . . , X d } and Y = { Y , . . . , Y d } aresets of Hermitian matices on C n with kX k ≤ and kY k ≤ . For anyorthonormal subset B = { b , . . . , b k } of C n , µ X ( B ) ≤ µ Y ( B ) + 4 d · dist ( X , Y ) . lmost commuting unitary matrices related to time reversal 28 Proof.
For any unit vector b we find (cid:12)(cid:12)(cid:10) X r b , b (cid:11) − (cid:10) Y r b , b (cid:11)(cid:12)(cid:12) ≤ (cid:13)(cid:13) X r − Y r (cid:13)(cid:13) ≤ k X r − Y r k , and (cid:12)(cid:12) h X r b , b i − h Y r b , b i (cid:12)(cid:12) ≤ |h X r b , b i − h Y r b , b i| ≤ k X r − Y r k , and so (cid:12)(cid:12) σ X r ( b ) − σ Y r ( b ) (cid:12)(cid:12) ≤ X , Y ) . Therefore (cid:12)(cid:12) σ X ( b j ) − σ X ( b j ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X r σ X r ( b j ) − X r σ Y r ( b j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d · dist ( X , Y ) , for all j . (cid:3) Lemma 7.2.
Suppose Y = { Y , . . . , Y d } is a set of commuting Hermit-ian matices on C n . There exists an orthonormal basis B = { b , . . . , b n } of C n for which µ Y ( B ) = 0 . Proof.
We take B to be an orthonormal basis of common eigenvectorsof the Y r and then notice (cid:10) Y r b j , b j (cid:11) − h Y r b j , b j i = 0 . (cid:3) Lemma 7.3.
Suppose W : C n → C n is an isometry. Let P = W W ∗ . Suppose X = { X , . . . , X d } is a set of Hermitian matices on C n with kX k ≤ and k [ X r , P ] k ≤ δ for all r . Suppose B = { b , . . . , b k } is anorthonormal set in C n . The orthonormal set W B = { W b , . . . , W b k } and the set W ∗ X W = { W ∗ X W, . . . , W ∗ X d W } satisfies µ X ( W B ) ≤ µ W ∗ X W ( B ) + 8 dδ. Proof.
Let Y r = P X r P + ( I − P ) X r ( I − P ). Then k X r − Y r k ≤ k P X r ( I − P ) k ≤ k [ P, X r ] k so dist ( X , Y ) ≤ δ and therefore, by Lemma 7.1, µ X ( W B ) ≤ µ Y ( W B ) + 8 dδ, where Y = { Y , Y , . . . , Y d } . For any b in C n we find h Y r W b , W b i = h W ∗ Y r W b , b i and (cid:10) Y r W b , W b (cid:11) = (cid:10) W ∗ Y r P W b , b (cid:11) = h W ∗ Y r W W ∗ Y r W b , b i = (cid:10) ( W ∗ Y r W ) b , b (cid:11) . lmost commuting unitary matrices related to time reversal 29 This implies µ W ∗ Y W ( B ) = µ Y ( W B ) . However, W ∗ Y r W = W ∗ X r W so µ W ∗ X W ( B ) = µ W ∗ Y W ( B ) = µ Y ( W B ) . (cid:3) Proposition 7.4.
Suppose P is a projection in M n ( C ) and ˆ X = { ˆ X , ˆ X , ˆ X , ˆ X } is a representation of T ′ and k [ P, ˆ X r ] k ≤ δ for all r . Let n be the rank of P and suppose W : C n → C n is any isometrywith W W ∗ = P . Let X r = W ∗ ˆ X r W and X = { X , X , X , X } . (1) kX k ≤ . (2) X = { X , X , X , X } is a representation of T ′ δ . (3) If there is a representation Y of T ′ with dist( X , Y ) ≤ ǫ thenthere is an orthogonal basis B of P C n with µ ˆ X ( B ) ≤ dδ + 4 dǫ. Proof. (1) The norm condition is easy, as it certainly is true for the ˆ X r .(2) For any r and s , k X r X s − X s X r k = (cid:13)(cid:13)(cid:13) W ∗ ˆ X r P ˆ X s W − W ∗ ˆ X s P ˆ X r W (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ˆ X r P ˆ X s − ˆ X s P ˆ X r (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ˆ X r P ˆ X s − ˆ X r ˆ X s P (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˆ X s ˆ X r P − ˆ X s P ˆ X r (cid:13)(cid:13)(cid:13) ≤ δ and, for ( r, s ) equal (1 ,
2) or (3 , (cid:13)(cid:13) X r + X s − I (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) W ∗ ˆ X r P ˆ X r W + W ∗ ˆ X s P ˆ X s W − W ∗ P W (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ˆ X r P ˆ X r + ˆ X s P ˆ X s − P (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) ˆ X r P ˆ X r − ˆ X r P (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˆ X s P ˆ X s − ˆ X s P (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ˆ X r P + ˆ X s P − P (cid:13)(cid:13)(cid:13) ≤ δ. (3) Follows from (1), (2) and Lemmas 7.1, 7.2, and 7.3. (cid:3) Definition 7.5.
If we had made a different choice of partial isometry, W , then the resulting 4-tuples would be unitarily equivalent to thefirst, since W ∗ W is a unitary. Therefore, the Bott index,Bott( X + iX , X + iX ) lmost commuting unitary matrices related to time reversal 30 does not depend on the choice of W . We therefore defineBott( P ; ˆ X , ˆ X , ˆ X , ˆ X ) = Bott( X + iX , X + iX )so long as δ is small enough to ensure a spectral gap in the Bott matrix B ( H , H , H ).In this form, depending on a projection almost commuting with com-muting matrices, our Bott index is essentially the same as the Chernnumber as calculated on finite samples, in [23].If the projection P and the ˆ X r are real, then we can select the isom-etry W to be real which means that the X r will be real.If the projection P and the ˆ X r are self-dual in M N ( C ), we canuse the spectral theorem for normal quaternionic matrices to find anisometry W with W W ∗ = P and W ∗ = − Z N W T Z N . This means X ♯r = − Z N W T ˆ X r W Z N = − W ∗ Z N ˆ X r Z N W = W ∗ ˆ X r W, so the X r are self-dual. We can unambiguously definePf − Bott( P ; ˆ X , ˆ X , ˆ X , ˆ X ) = Pf − Bott( X + iX , X + iX ) . We finish then with corollaries to the theorems in § Theorem 7.6.
For every ǫ > , there is a δ in (0 , ) so that, for ev-ery -tuple of commuting Hermitian matrices ˆ X , ˆ X , ˆ X , ˆ X in M n ( C ) with ˆ X + ˆ X = ˆ X + ˆ X = I, if P is a projection with k [ P, ˆ X r ] k ≤ δ for all r , and if Bott( P ; ˆ X , ˆ X , ˆ X , ˆ X ) = 0 , then there there is an orthonormal basis b , . . . , b n of the subspace P C n so that D ˆ X r b j , b j E − D ˆ X r b j , b j E ≤ ǫ for all r and all j . Theorem 7.7.
For every ǫ > , there is a δ > so that, for every -tuple of commuting Hermitian matrices ˆ X , ˆ X , ˆ X , ˆ X in M n ( R ) with ˆ X + ˆ X = ˆ X + ˆ X = I, if P is a real projection with k [ P, ˆ X r ] k ≤ δ for all r , then there there isan orthonomal basis b , . . . , b n of the subspace P R n so that D ˆ X r b j , b j E − D ˆ X r b j , b j E ≤ ǫ lmost commuting unitary matrices related to time reversal 31 for all r and j . Theorem 7.8.
For every ǫ > , there is a δ in (0 , ) so that, for every -tuple of commuting Hermitian, self-dual matrices ˆ X , ˆ X , ˆ X , ˆ X in M N ( C ) with ˆ X + ˆ X = ˆ X + ˆ X = I, if P is a self-dual projection with k [ P, ˆ X r ] k ≤ δ for all r , and if Pf − Bott( P ; ˆ X , ˆ X , ˆ X , ˆ X ) = 1 , then there are vectors b , . . . , b N so that b , . . . , b N , T b , . . . , T b N is an orthonormal basis of the subspace P C N and D ˆ X r b j , b j E − D ˆ X r b j , b j E ≤ ǫ for all r and all j . Discussion
Our results on localization have very minimal assumptions. We donot require that the projection P arises from a gap in the specrtrum ofthe Hamiltonian. We also allow that the distribution of sites within thetorus can be very irregular. We pay for this with very weak localizationin the conclusion. In contrast, one can study specific models and findexponential decay in a basis made from time-reversal pairs, see [25].Our results might have applications in signal processing in areas suchas hypercomplex processes [27] and blind source separation [2, 28]. Acknowledgments
We thank Matthew Hastings for his suggestions regarding the presentarticle and for his role in laying out a program relating the theory ofalmost commuting matrices to condensed matter physics.This research was supported by the Danish National Research Foun-dation (DNRF) through the Centre for Symmetry and Deformation.
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