LLOCAL MATRIX HOMOTOPIES AND SOFT TORI
TERRY A. LORING AND FREDY VIDES
Abstract.
We present solutions to local connectivity problems in matrix rep-resentations of the form C ([ − , N ) → A n,ε ← C ε ( T ) for any ε ∈ [0 , n ≥
1, where A n,ε ⊆ M n and where C ε ( T ) denotes the Soft Torus . We solve the connectivity problems by introducing the so calledtoroidal matrix links, which can be interpreted as normal contractive matrixanalogies of free homotopies in differential algebraic topology.In order to deal with the locality constraints, we have combined some tech-niques introduced in this document with some techniques from matrix geom-etry, combinatorial optimization, classification and representation theory ofC ∗ -algebras. Introduction
In this document we study the solvability of some local connectivity problemsvia constrained normal matrix homotopies in C ∗ -representations of the form(1.1) C ( T N ) −→ M n , for a fixed but arbitrary integer N ≥ n ≥
1. In particularwe study local normal matrix homotopies which preserve commutativity and alsosatisfy some additional constraints, like being rectifiable or piecewise analytic.We build on some homotopic techniques introduced initially by Bratteli, Elliot,Evans and Kishimoto in [3] and generalized by Lin in [18] and [22]. We combinethe homotopic techniques with some techniques introduced here and some othertechniques from matrix geometry and noncommutative topology developed by Lor-ing [24, 27], Shulman [27], Bhatia [1], Chu [7], Brockett [4], Choi [6, 5], Effros [5],Exel [10], Eilers [10], Elsner [11], Pryde [30, 29], McIntosh [29] and Ricker [29],to construct the so called toroidal matrix links , which we use to obtain the maintheorems presented in section §
4, and which consist on local connectivity results inmatrix representations of the form 1.1 and also of the form(1.2) C ( T N ) −→ M n ←− C ([ − , N ) . Toroidal matrix links can be interpreted as noncommutative analogies of freehomotopies in algebraic topology and topological deformation theory, they are in-troduced in section § § C ( T ) → M n . Date : November 14, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Matrix homotopy, relative lifting problems, matrix representation,noncommutative semialgebraic sets, K-theory, amenable C*-algebra, joint spectrum. a r X i v : . [ m a t h . OA ] J un TERRY A. LORING AND FREDY VIDES
Given δ >
0, a function ε : R → R +0 and two matrices x, y in a set S ⊆ M n such that (cid:107) x − y (cid:107) ≤ δ , by a ε ( δ ) -local matrix homotopy between x and y , wemean a matrix path X ∈ C ([0 , , M n ) such that X = x , X = y , X t ∈ S and (cid:107) X t − y (cid:107) ≤ ε ( δ ) for each t ∈ [0 , x (cid:32) ε y to denote that there is a ε -localmatrix homotopy betweeen x and y .The motivation and inspiration to study local normal matrix homotopies whichpreserve commutativity in C ∗ -representations of the form 1.1 and 1.2, came frommathematical physics [15, §
3] and matrix approximation theory [8].The problems from mathematical physics which motivated this study are inversespectral problems, which consist on finding for a certain set of matrices X , . . . , X N which approximately satisfy a set of polynomial constraints R ( x , . . . , x N ) on N NC-variables, a set of nearby matrices ˜ X , . . . , ˜ X N which approximate X , . . . , X N and exactly satisfy the constraints R ( x , . . . , x N ). The problems from matrix ap-proximation theory that we considered for this study, are of the type that can bereduced to the study of the solvability conditions for approximate and exact jointdiagonalization problems for N -tuples of normal matrix contractions.Since the problems which motivated the research reported in this document canbe restated in terms of the study local piecewise analytic connectivity in matrix rep-resentations of the form C ε ( T ) → M n ← C ( T N ) and C ε ( T ) → M n ← C ([ − , N ),we studied several variations of problems of the form. Problem 1 (Lifted connectivity problem) . Given ε > , is there δ > such thatthe following conditions hold? For any integer n ≥ , some prescribed sequenceof linear compressions κ n : M mn → M n for some m ≥ , and any two familiesof N pairwise commuting normal contractions X , . . . , X N and Y , . . . , Y N in M n which satisfy the constraints (cid:107) X j − Y j (cid:107) ≤ δ , ≤ j ≤ N , there are two familiesof N pairwise commuting normal contractions ˜ X , . . . , ˜ X N and ˜ Y , . . . , ˜ Y N in M mn which satisfy the relations: κ ( ˜ X j ) = X j , κ ( ˜ Y j ) = Y j and (cid:107) ˜ X j − ˜ Y j (cid:107) ≤ ε , ≤ j ≤ N . Moreover, there are N peicewise analytic ε -local homotopies of normalcontractions X , . . . , X N ∈ C ([0 , , M mn ) between the corresponding pairs ˜ X j , ˜ Y j in M mn , which satisfy the relations X jt X kt = X kt X jt , for each ≤ j, k ≤ N andeach ≤ t ≤ . By solving problem P. δ -close N -tuples of pairwise commuting normal contractions X , . . . , X N and Y , . . . ,Y N in M n , which was the main motivation of the research reported here. We alsoobtained some results concerning to the geometric structure of the joint spectra (inthe sense of [29]) of the N -tuples.For a given δ >
0, the study of the solvability conditions of problems of the form P. C ( T N ) → A := C ∗ ( U , . . . , U N ) ⊆ M n and C ( T N ) → A := C ∗ ( V , . . . , V N ) ⊆ M n , where U , . . . , U N , V , . . . , V N ∈ U ( n ) are pairwisecommuting unitary matrices such that (cid:107) U j − V j (cid:107) ≤ δ . By local deformations wemean a family { A t } t ∈ [0 , ⊆ M n of abelian C ∗ -algebras, with A t := C ∗ ( X t , . . . , X Nt )and where X t , . . . , X Nt ∈ C ([0 , , U ( n )) are ε ( δ )-local matrix homotopies between U , . . . , U N and V , . . . , V N for some function ε : R → R +0 .The main results are presented in §
4, in section § § OCAL MATRIX HOMOTOPIES 3 technique via matrix homotopy lifting and in section § Preliminaries and Notation
Matrix Sets and Operations.
Given two elements x, y in a C ∗ -algebra A ,we will write [ x, y ] and Ad[ x ]( y ) to denote the operations [ x, y ] := xy − yx andAd[ x ]( y ) := xyx ∗ .Given any C ∗ -algebra A and any element x in M n ( A ), we will denote by diag n [ x ]the operation defined by the expression M n ( A ) → M n ( A ) x (cid:55)→ diag n [ x ] x x · · · x n x x · · · x n ... ... . . . ... x n x n · · · x nn (cid:55)→ x · · · x · · · · · · x nn . Given a C ∗ -algebra A , we write N ( A ), H ( A ) and U ( A ) to denote the sets ofnormal, hermitian and unitary elements in A respectively. We will write N ( n ), H ( n ) and U ( n ) instead of N ( M n ), H ( M n ) and U ( M n ). A normal element u in aC ∗ -algebra A is called a partial unitary if the element uu ∗ = p is an otrhogonalprojection in A , i.e. p satisfies the relations p = p ∗ = p , we denote by PU ( A ) theset of partial unitaries in A and we write PU ( n ) instead of PU ( M n ).We write I , J , T and D to denote the sets I := [0 , J = [ − , T := { z ∈ C | | z | = 1 } and D := { z ∈ C | | z | ≤ } . For some arbitrary matrix set S ⊆ M n and some arbitrary compact set X ⊂ C , we will write S ( X ) to denote the subset ofelements in S described by the expression,(2.1) S ( X ) := { x ∈ S | σ ( x ) ⊆ X } , for instance we can write N ( n )( D ) to denote the set of nomal contractions. Wewill denote by M ∞ the C ∗ -algebra described by(2.2) M ∞ := (cid:91) n ∈ Z + M n (cid:107)·(cid:107) . In this document we write n to denote the identity matrix in M n . The symbol N n will be used to denote the diagonal matrices(2.3) N n := diag [ n, n − , . . . , , . We will write Ω n and Σ n to denote the unitary matrices defined byΩ n := e πin N n = diag (cid:104) , e πi ( n − n , . . . , e πin , e πin (cid:105) (2.4)and Σ n := (cid:18) n − (cid:19) . (2.5) TERRY A. LORING AND FREDY VIDES
Remark 2.1.
The unitary matrices Ω n and Σ n are related, by the equation Ω n = F ∗ n Σ n F n , where F N := (cid:16) √ N e πi ( j − k − N (cid:17) ≤ j,k ≤ N is the discrete Fourier transform (DFT)unitary matrix. Given an abstract object (group or C ∗ -algebra) A we write A ∗ N to denote theoperation consisting on taking the free product of N copies of A . Definition 2.1 (Local preservers) . Given a linear mapping K : M N → M n , with n ≤ N , and given a set S ⊆ M n , we say that K locally preserves S if there is T ⊆ M N such that K ( T ) ⊆ S , if in particular K ( T ) ⊆ N ( n ) we say that K locallypreserves normality . Example 2.1.
The linear compression κ : M n → M n defined by κ : (cid:18) x x x x (cid:19) (cid:55)→ x locally preserves normality with respect to the set T := { X ∈ M n | x ∈ N ( n ) } . Example 2.2.
The linear map φ : M n → M n , x (cid:55)→ D x with n ≥ and D = n diag [1 , . . . , n ] , locally preserves commutativity with respect to the set C ∗ ( D ) . Joint Spectral Variation.
Clifford Operators.
Using the same notation as Pryde in [30], let R ( N ) denotethe Clifford algebra over R with generators e , . . . , e N and relations e i e j = − e j e i for i (cid:54) = j and e i = −
1. Then R ( N ) is an associative algebra of dimension 2 N . Let S ( N )denote the set P ( { , . . . , N } ). Then the elements e S = e s · · · e s k form a basiswhen S = { s , . . . , s k } and 1 ≤ s < · · · < s k ≤ N . Elements of R ( N ) are denotedby λ = (cid:80) S λ S e S where λ S ∈ R . Under the inner product (cid:104) µ, = (cid:105) λ (cid:80) S λ S µ S , R ( N ) becomes a Hilbert space with orthonormal basis { e S } .The Clifford operator of N elements X , . . . , X N ∈ M n is the operator definedin M n ⊗ R ( N ) by Cliff( X , . . . , X N ) := i N (cid:88) j =1 X j ⊗ e j . Each element T = (cid:80) S T S ⊗ e S ∈ M n ⊗ R ( N ) acts on elements x = (cid:80) S x S ⊗ e S ∈ C n ⊗ R ( N ) by T ( x ) := (cid:80) S,S (cid:48) T s ( x S (cid:48) ) ⊗ e S e S (cid:48) . So Cliff( X , . . . , X N ) ∈ M n ⊗ R ( N ) ⊆ L ( C n ⊗ R ( N ) ). By (cid:107) Cliff( X , . . . , X N ) (cid:107) we will mean the operator norm ofCliff( X , . . . , X N ) as an element of L ( C n ⊗ R ( N ) ). As observed by Elsner in [11,5.2] we have that(2.6) (cid:107) Cliff( X , . . . , X N ) (cid:107) ≤ N (cid:88) j =1 (cid:107) X j (cid:107) . Joint Spectral Matchings.
It is often convenient to have N -tuples (or 2 N -tuples) of matrices with real spectra. For this purpose we use the following con-struction, initiated by McIntosh and Pryde. If X = ( X , . . . , X N ) is a N -tuple of n by n matrices then we can always decompose X j in the form X j = X j + iX j wherethe X kj all have real spectra. We write π ( X ) := ( X , . . . , X N , X , . . . , X N ) andcall π ( X ) a partition of X . If the X kj all commute we say that π ( X ) is a commuting OCAL MATRIX HOMOTOPIES 5 partition, and if the X kj are simultaneously triangularizable π ( X ) is a triangular-izable partition. If the X kj are all semisimple (diagonalizable) then π ( X ) is calleda semisimple partition.We say that N normal matrices X , . . . , X N ∈ M n are simultaneously diagonal-izable if there is a unitary matrix Q ∈ M n such that Q ∗ X j Q is diagonal for each j = 1 , . . . , N . In this case, for 1 ≤ k ≤ n , let Λ ( k ) ( X j ) := ( Q ∗ X j Q ) kk the ( k, k )element of Q ∗ X j Q , and set Λ ( k ) ( X , . . . , X N ) := (Λ ( k ) ( X ) , . . . , Λ ( k ) ( X N )) ∈ C N .The set Λ( X , . . . , X N ) := { Λ ( k ) ( X , . . . , X N ) } ≤ k ≤ N is called the joint spectrum of X , . . . , X N . We will write Λ( X j ) to denote the j -component of Λ( X , . . . , X N ), in other words we will have thatΛ( X j ) = diag (cid:104) Λ (1) ( X j ) , . . . , Λ ( N ) ( X j ) (cid:105) . The following theorem was proved in McIntosh, Pryde and Ricker [29].
Theorem 2.1 (McIntosh, Pryde and Ricker) . Let X = ( X , . . . , X N ) and Y =( Y , . . . , Y N ) be N -tuples of commuting n by n normal matrices. There exists apermutation τ of the index set { , . . . , n } such that (2.7) (cid:107) Λ ( k ) ( X , . . . , X N ) − Λ ( τ ( k )) ( Y , . . . , Y N ) (cid:107) ≤ e N, (cid:107) Cliff( X − Y , . . . , X N − Y N ) (cid:107) for all k ∈ { , . . . , n } . In this theorem, e N, is an explicit constant depending only on N defined in [29,(2.4)].2.3. Amenable C ∗ -algebras and Bott elements. The following lemma hasbeen proved by H. Lin in [19].
Lemma 2.1 (H. Lin.) . For any ε > and d > , there exists δ > satisfying thefollowing: Suppose that A is a unital C ∗ -algebra and u ∈ A is a unitary such that T \ σ ( u ) contains an arc with length d . Suppose that a ∈ A with (cid:107) a (cid:107) ≤ such that (cid:107) ua − au (cid:107) < δ. Then there is a self-adjoint element h ∈ A such that u = e ih , (cid:107) ha − ah (cid:107) < ε and (cid:107) e ith a − ae ith (cid:107) < ε for all t ∈ I . If, furthermore, a = p is a projection, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pup − p + ∞ (cid:88) n =1 ( iphp ) n n ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε. The following lemma was proved by H. Lin in [22] using L.2.1, since for anyinteger n ≥ u ∈ U ( n ), we will have that T \ σ ( u ) contains an arc withlength at least 2 π/n . Lemma 2.2 (H. Lin.) . Let ε > , n ≥ be an integer and M > . There exists δ > satisfying the following: For any finite set F ⊂ M n with (cid:107) a (cid:107) ≤ M for all a ∈ F , and a unitary u ∈ M n such that (cid:107) ua − au (cid:107) < δ for all a ∈ F , TERRY A. LORING AND FREDY VIDES there exists a continuous path of unitaries { u ( t ) } t ∈ I ⊂ M n with u (0) = u and u (1) = n such that (cid:107) u ( t ) a − au ( t ) (cid:107) < ε for all a ∈ F . Morover,
Length ( { u ( t ) } ) ≤ π. Definition 2.2. ( The obstruction
Bott ( u, v ) . ) Given two unitaries in a K -simple real rank zero C ∗ -algebra A that almost commute, the obstruction Bott ( u, v ) is the Bott element associated to the two unitaries as defined by Loring in [24] . Itis defined whenever (cid:107) uv − vu (cid:107) ≤ ν , where ν is a universal constant. It is definedas the K -class Bott ( u, v ) = [ χ [1 / , ∞ ) ( e ( u, v ))] − (cid:20)(cid:18) (cid:19)(cid:21) , where e ( u, v ) is a self-adjoint element of M ( A ) of the form e ( u, v ) = (cid:18) f ( v ) h ( v ) u + g ( v ) u ∗ h ( v ) + g ( v ) 1 − f ( v ) (cid:19) , where f , g , h are certain universal real-valued continuous functions on T . For details on the subject of K -theory for C ∗ -algebras the reader is referred to[31]. As observed by Bratteli, Elliot, Evans and Kishimoto in [3], given a pair u, v ∈ U ( A ) we have that the obstruction Bott ( u, v ) needs to vanish in order to beable to solve the problem uvu ∗ (cid:32) ε ( δ ) v by deforming u ∈ U ( A ) to 1 continuouslyin U ( A ), when (cid:107) uv − vu (cid:107) ≤ δ .3. Matrix Varieties and Toroidal Matrix Links
Let us denote by H a universal separable Hilbert space, by B ( H ) the C ∗ -algebraof bounded operators on H , and for any given S ⊆ B ( H ) let us denote by B r ( S )the closed r -ball in S defined by B r ( S ) := { x ∈ S |(cid:107) x (cid:107) ≤ r } .Given some N ∈ Z + and a set R ( S ) = R ( y , . . . , y N ) of normed polynomialrelations on the N -set S := { y , . . . , y N } of NC-variables, we will call the set Z [ R ]described by(3.1) Z [ R ] := { x , . . . , x N |R ( x , . . . , x N ) } with x , . . . , x N ∈ B ( B ( H )), a noncommutative semialgebraic set. Example 3.1.
As an example of normed
N C -polynomial relations we can considerthe set R ( x, y ) := {(cid:107) x − (cid:107) ≤ − , (cid:107) y − (cid:107) ≤ − , (cid:107) xy − yx (cid:107) ≤ , xx ∗ = x ∗ x =1 , yy ∗ = y ∗ y = 1 } . Given a NC-semialgebraic set Z [ R ], we will use the symbol EZ [ R ] to denote theuniversal C ∗ -algebra(3.2) EZ [ R ] := C ∗ (cid:104) x , . . . , x N |R ( x , . . . , x N ) (cid:105) , which we call the environment C ∗ -algebra of Z [ R ]. For details on universal C ∗ -algebras described in terms of generators and relations the reader is referred to[26]. OCAL MATRIX HOMOTOPIES 7
Definition 3.1 (Semialgebraic Matrix Varieties) . Given J ∈ Z + , a system of J polynomials p , . . . , p J ∈ Π (cid:104) N (cid:105) = C (cid:104) x , . . . , x N (cid:105) in N NC-variables x , . . . , x N ∈ Π (cid:104) N (cid:105) and a real number ε ≥ , a particular matrix representation of the noncom-mutative semialgebraic set Z ε,n ( p , . . . , p J ) described by (3.3) Z ε,n ( p , . . . , p J ) := { X , . . . , X N ∈ M n | (cid:107) p j ( X , . . . , X N ) (cid:107) ≤ ε, ≤ j ≤ J } , will be called a ε, n -semialgebraic matrix variety ( ε, n -SMV), if ε = 0 we canrefer to the set as a matrix variety . Example 3.2.
As a first example, we will have that the set Z n := { X ∈ M n | N n X − X N n = 0 } is a matrix variety defined by the set with one N C -polynomial relation { N n X − X N n = 0 } . If for some δ > , we set now Z n,δ := { X ∈ M n |(cid:107) [ N n , X ] (cid:107) ≤ δ } , the set Z n,δ is a matrix semialgebraic variety defined by the set with one normed N C -polynomial relation {(cid:107) N n X − X N n (cid:107) ≤ δ } . Example 3.3.
Other example of a matrix semialgebraic variety, that has beenuseful to understand the geometric nature of the problems solved in this document,is described by the matrix set
Iso δ ( x, y ) , defined for some given δ ≥ and any twonormal contractions x and y in M n , by the expression Iso δ ( x, y ) := ( z, w ) ∈ N ( n )( D ) × U ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:107) xw − wz (cid:107) = 0 (cid:107) [ z, y ] (cid:107) = 0 (cid:107) z − y (cid:107) ≤ δ . Toroidal Matrix Links.
Finsler manifolds, matrix paths and toroidal matrix links.
Definition 3.2 (Finsler manifold) . A Finsler manifold is a pair ( M, F ) where M is a manifold and F : T M → [0 , ∞ ) is a function (called a Finsler norm) such that • F is smooth on T M \{ } = (cid:83) x ∈ M { T x M \{ }} , • F ( v ) ≥ with equality if and only if v = 0 , • F ( λv ) = λF ( v ) for all λ ≥ , • F ( v + w ) ≤ F ( v ) + F ( w ) for all w at the same tangent space with v . Given a Finsler manifold (
M, F ), the length of any rectifiable curve γ : [ a, b ] → M is given by the length functional L [ γ ] = (cid:90) ba F ( γ ( t ) , ∂ t γ ( t )) dt, where F ( x, · ) is the Finsler norm on each tangent space T x M .The pair ( N , (cid:107) · (cid:107) ) is a Finsler manifold, where N denotes the set of normalmatrices N (of any size) and (cid:107) · (cid:107) denotes the operator norm. Definition 3.3 (Matrix path curvature) . Given a piecewise- C matrix path γ :[0 , → N , we define its curvature κ [ γ ] to be κ [ γ ] := 1 (cid:107) ∂ t γ ( t ) (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) ∂ t (cid:18) ∂ t γ ( t ) (cid:107) ∂ t γ ( t ) (cid:107) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) . Definition 3.4 (Matrix flows) . Given n ≥ , a mapping φ : R + × M n → M n , ( t, x ) (cid:55)→ x t will be called a matrix flow in this document. If we have in additionthat σ ( x t ) = σ ( x s ) for every t, s ≥ , we say that the matrix flow is isospectral. TERRY A. LORING AND FREDY VIDES
Definition 3.5 (interpolating path) . Given two matrices x and y in M n and amatrix flow φ : I × M n → M n such that φ ( x ) = x and φ ( x ) = y , we say that thecorresponding path { x t } t ∈ I := { φ t ( x ) } t ∈ I ⊆ M n is a solvent path for the interpola-tion problem x (cid:32) y . Definition 3.6 ( (cid:126) operation) . Given two matrix paths
X, Y ∈ C ([0 , , M n ) wewrite X (cid:126) Y to denote the concatenation of X and Y , which is the matrix pathdefined in terms of X and Y by the expression, X (cid:126) Y s := (cid:26) X s , ≤ s ≤ ,Y s − , ≤ s ≤ . Definition 3.7. ( (cid:96) (cid:107)·(cid:107) ) Given a matrix path { x t } t ∈ I in M n we will write (cid:96) (cid:107)·(cid:107) ( x t ) todenote the length of { x t } t ∈ I with respect to the operator norm which is defined bythe expression (cid:96) (cid:107)·(cid:107) ( x t ) := sup m − (cid:88) k =0 (cid:107) x t k +1 − x t k (cid:107) , where the supremum is taken over all partitions of I as t < . . . < t m = b . Ifthe function x ∈ C ( I , M n ) is a piecewise C function, then (cid:96) (cid:107)·(cid:107) ( x t ) = (cid:90) I (cid:107) ∂ t x t (cid:107) dt. Definition 3.8. ( (cid:107) · (cid:107) -flatness ) A set S of M n is said to be (cid:107) · (cid:107) -flat if any twopoints x, y ∈ S can be connected by a path { x t } t ∈ I ⊆ S such that (cid:96) (cid:107)·(cid:107) ( x t ) = (cid:107) x − y (cid:107) . Definition 3.9 (Toroidal matrix link) . Given any two normal contractions x, y in M n , a toroidal matrix link is any piecewise analytic normal path x t := K [ T t ( l ( x ))] induced by a locally normal piecewise analytic matrix flow T : I × M N → M N with N ≥ n , together with a locally normal compression K : M N → M n with relativelifting map l : M n → M N , which satisfy the interpolating conditions K [ T ( l ( x ))] = x and K [ T ( l ( x ))] = y together with the constraints (cid:107) K [ T t ( l ( x ))] (cid:107) ≤ for each t ∈ I . Remark 3.1.
In the particular case where [ K ( T t ( l ( x ))) , K ( T t ( l ( y )))] = 0 for each t ∈ I , whenever [ x, y ] = 0 , we call T a toral matrix link. Remark 3.2.
The curved nature of the matrix varieties ( as Finsler sub-manifolds of N ) whose local connectivity is studied in this document, induces anobstruction to local connectivity via entirely flat toroidal matrix links in general.The toroidal matrix links T ⊂ C ([0 , , N ) we have used to solve the connectivityproblems which motivated this study satisfy the constraint ≤ κ [ T ] ≤ (cid:96) (cid:107)·(cid:107) ( T ) , ∀ T ∈ T . Embedded matrix flows in solid tori.
Given some fixed but arbitrary W ∈ U ( n ), using the operation diag n : M n → M n one can define the mapping D : U ( n ) × M n → D , determined by the expression. U ( n ) × M n → D (3.4) ( W, x ) (cid:55)→ D T [ W ]( x )(3.5) ( W, x ) (cid:55)→ { (diag n [ W xW ∗ ]) k,k } ≤ k ≤ n (3.6)It is clear that diag [ D T [ W ]( x )] = diag n [ W xW ∗ ] and that diag [ D T [ n ]( x )] =diag n [ x ], because of this when W = n we will write D ( x ) instead of D T [ n ]( x ). OCAL MATRIX HOMOTOPIES 9
Given a matrix flow I × N ( n )( D ) → N ( n )( D ) , ( t, x ) (cid:55)→ X t ( x ), one can iden-tify X with the set of flow lines in D × T determined by { ( D ( X t ( x )) , e πit ) } t ∈ I .The geometric picture determined by the mapping cylinder N ( n )( D ) × I → D × T , ( x, t ) (cid:55)→ ( D ( X t ( x )) , e πit ) will be called the embedded matrix mapping cylinderrelative to the flow X . We can think of the embedded matrix mapping cylinderin topological terms as a deformation described by the expression D X,Z , which isdefined as(3.7) D X,Z [ Z × I ] := ( Z × I ) (cid:116) Z Z (cid:32) X Z , where Z and Z are some prescribed matrix varieties such that x ∈ Z and X ( x ) ∈ Z . Example 3.4 (Graphical example in M ) . Let us set ˆ u := e πi f ( N ) where f ∈ C ( I , I ) . For some prescribed W ∈ U (3) , we can obtain a graphical exam-ple of a particular geometric picture of the computation of the embedded matrixmapping cylinder relative to the interpolating flow U which solves the problem ˆ u (cid:32) W ˆ u W ∗ .Let us set Z := { z ∈ U (3) | [ˆ u , z ] = 0 } ,Z := { z ∈ U (3) | [ W ˆ u W ∗ , z ] = 0 } . Using projective methods, we can trace specific flow lines along the matrix flowscorresponding to the dynamical deformation D U ,Z [ Z × I ] , which solve the interpo-lation problem ˆ u (cid:32) W ˆ u W ∗ .A particular (approximate) geometric picture of the matrix deformation inducedby the toral matrix link { U t } t ∈ I in M , projected in D × T for each t ∈ I via D T ( U t ) is presented in figures F.1-F.3. Figure 1.
Projected matrix mapping cylinder corresponding tothe path U [0 , ] (ˆ u ) in M . Alternative methods to trace particular flow lines on mapping cylinders can beotained using matrix homotopies, this can be done using similar methods to theones implemented in [7] . Environment algebras.Definition 3.10 (Environment algebra (of a matrix algebra)) . Given a mat-rixalgebra A ⊆ M n , a universal C ∗ -algebra E A := C ∗ (cid:104) x , . . . , x m |R ( x , . . . , x m ) (cid:105) for Figure 2.
Projected matrix mapping cylinder corresponding tothe path U I (ˆ u ) in M . Figure 3.
Embedded matrix mapping cylinder corresponding tothe path U I (ˆ u ) in M . which there is a matrix representation E A (cid:16) E A ⊆ M n such that A ⊆ E A , will becalled an environment algebra for A . Let us consider the universal C ∗ -algebras C ( J ), C ( T ), C ( T ) ∗ C C ( T ), C δ ( T ), C δ ( J × T ) and C ∗ ε (cid:104) Z / × Z (cid:105) , defined in terms of generators and relations by theexpressions. C ( J ) := C ∗ (cid:10) u (cid:12)(cid:12) h ∗ = h, (cid:107) h (cid:107) ≤ (cid:11) C ( T ) := C ∗ (cid:10) u (cid:12)(cid:12) uu ∗ = u ∗ u = 1 (cid:11) C ( T ) ∗ C C ( T ) := C ∗ (cid:28) u, v (cid:12)(cid:12)(cid:12)(cid:12) uu ∗ = u ∗ u = 1 ,vv ∗ = v ∗ v = 1 (cid:29) C δ ( T ) := C ∗ (cid:42) u, v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) uu ∗ = u ∗ u = 1 ,vv ∗ = v ∗ v = 1 , (cid:107) uv − vu (cid:107) ≤ δ (cid:43) OCAL MATRIX HOMOTOPIES 11 C δ ( J × T ) := C ∗ (cid:42) h, u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ∗ = h, (cid:107) h (cid:107) ≤ uu ∗ = u ∗ u = 1 , (cid:107) hu − uh (cid:107) ≤ δ (cid:43) C ∗ ε (cid:104) Z / × Z (cid:105) := C ∗ (cid:42) u, v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) uu ∗ = u ∗ u = u = 1 ,vv ∗ = v ∗ v = 1 , (cid:107) uv − vu (cid:107) ≤ ε (cid:43) . Let us consider now a local matrix representation result that we will use later inthe construction of particular representation schemes.
Lemma 3.1.
For every integer n ≥ , there are s , u n , v n ∈ U ( M ∞ ) such that thediagram C ( T ) ∗ (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) C ∗ (cid:104) ( Z /n ) ∗ (cid:105) (cid:47) (cid:47) (cid:47) (cid:47) C ∗ n ( u n , v n ) C ∗ (cid:104) Z /n ∗ Z / (cid:105) (cid:47) (cid:47) (cid:47) (cid:47) C ∗ n ( s , v n ) M n commutes, where s ∈ H ( n ) , u n and v n are unitary elements in M n .Proof. Since we have that C ( T ) ∗ (cid:39) C ∗ (cid:104) F (cid:105) (cid:39) C ∗ ( Z ∗ ), by universality of theC ∗ -representa-tions C ∗ ( Z ∗ ) (cid:39) C ∗ (cid:28) u, v (cid:12)(cid:12)(cid:12)(cid:12) uu ∗ = u ∗ u = ,vv ∗ = v ∗ v = (cid:29) C ∗ (( Z /n ) ∗ ) (cid:39) C ∗ (cid:42) u, v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) uu ∗ = u ∗ u = ,vv ∗ = v ∗ v = ,u n = v n = (cid:43) C ∗ ( Z /n ∗ Z / (cid:39) C ∗ (cid:42) u, v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) uu ∗ = u ∗ u = ,vv ∗ = v ∗ v = ,u n = v = (cid:43) , and by the structural properties of M n , it is enough to find for any n ∈ Z + , up tounitary congruence in M n , three unitaries s , u n , v n ∈ U ( n ) such that C ∗ ( s , v n ) = M n = C ∗ ( u n , v n ) and u nn = v nn = s = n , this can be done by taking for any n ∈ Z + the orthogonal projection p := diag [1 , , . . . , ∈ H ( n ) and the matrix s = − p ∈ H ( n ), setting u n := Ω n and v n := Σ n for n ≥ u = v = 1for n = 1, by functional calculus and direct computations it is easy to verify that s , u n , v n ∈ U ( n ) for every n ∈ Z + , and that s = s ∗ , it is also easy to verifythat the system of matrix units { e i,j,n } ≤ i,j ≤ n and u n can be expressed as wordsin C ∗ ( s , v n ) for every n ∈ Z + , it is also clear that p = e , ,n and hence, s can be written as linear combinations of words in C ∗ ( u n , v n ), we will then havethat C ∗ (cid:104) Z /n ∗ Z / (cid:105) (cid:16) C ∗ ( v n , s ) and C ∗ (cid:104) Z /n ∗ (cid:105) (cid:16) C ∗ ( u n , v n ) by the universalproperties of C ∗ (cid:104) Z / ∗ Z /n (cid:105) and C ∗ (cid:104) Z /n ∗ (cid:105) respectively, since it can be easilyverified that u nn = v nn = s = n , from these facts and the universal property of C ( T ) ∗ (cid:39) C ∗ (cid:104) F (cid:105) (cid:39) C ∗ (cid:104) Z ∗ (cid:105) , theresult follows. (cid:3) Remark 3.3.
It can be seen that for any matrix C ∗ -subalgebra A ⊆ M n , thereis δ > such that both C ( T ) ∗ C C ( T ) and C δ ( T ) are environment algebras of A . It can also be seen that for any abelian C ∗ -subalgebra D ⊆ M n , C ( T ) is anenvironment algebra of D . Local Matrix Connectivity
Topologically controlled linear algebra and Soft Tori.Definition 4.1 (Controlled sets of matrix functions) . Given δ > , a function ε : R → R +0 , a finite set of functions F ⊆ C ( T , D ) and two unitary matrices u, v ∈ M n such that (cid:107) uv − vu (cid:107) ≤ δ , we say that the set F is δ -controlled by Ad[ v ] if the diagram, C ∗ ( u, v ) C ∗ ( u ) (cid:111) (cid:111) Ad[ v ] (cid:15) (cid:15) { u } Ad[ v ] (cid:15) (cid:15) ı (cid:111) (cid:111) f ≈ ε ( δ ) (cid:37) (cid:37) C ∗ ( vuv ∗ ) (cid:102) (cid:102) { vuv ∗ } ı (cid:111) (cid:111) f (cid:47) (cid:47) N ( n )( D ) commutes up to an error ε ( δ ) for each f ∈ F . Remark 4.1.
The C ∗ -homomorphism C δ ( T ) → C ∗ ( u, v ) allows us to see thatthe Soft Torus C δ ( T ) provides an environment algebra for any δ -controlled set ofmatrix functions. Lemma 4.1 (Existence of isospectral approximants) . Given ε > there is δ > such that, for any families of N pairwise commuting normal matrices x , . . . , x N and y , . . . , y N which satisfy the constraints (cid:107) x j − y j (cid:107) ≤ δ for each ≤ j ≤ N ,there is a C ∗ -homomorphism Ψ such that σ (Ψ( x j )) = σ ( x j ) , [Ψ( x j ) , y j ] = 0 and max {(cid:107) Ψ( x j ) − y j (cid:107) , (cid:107) Ψ( x j ) − x j (cid:107)} ≤ ε , for each ≤ j ≤ N .Proof. By changing basis if necessary, we can assume that y , . . . , y N are diagonalmatrices. From T.2.1 we will have that there is a permutation τ of the index set { , . . . , n } such that for each 1 ≤ k ≤ n we have that | Λ ( k ) ( x j ) − Λ ( τ ( k )) ( y j ) | ≤ (cid:107) Λ ( k ) ( x , . . . , x N ) − Λ ( τ ( k )) ( y , . . . , y N ) (cid:107)≤ e N, (cid:107) Cliff( x − y , . . . , x N − y N ) (cid:107) . (4.1)Using 2.6 and as a consequence of 4.1 we can find a permutation matrix T ∈ U ( n )such that (cid:107)T ∗ diag [Λ( x j )] T − diag [Λ( y j )] (cid:107) ≤ e N, (cid:107) Cliff( x − y , . . . , x N − y N ) (cid:107)≤ e N, N δ, ≤ j ≤ N. (4.2)Let us set c N := e N, N . For the matrices x , . . . , x N there is a unitary jointdiagonalizer W ∈ M n such that W diag [Λ( x j )] W ∗ = x j , 1 ≤ j ≤ N , (cid:107) W diag [Λ( x j )] W ∗ − T ∗ diag [Λ( x j )] T (cid:107) ≤ (cid:107) W diag [Λ( x j )] W ∗ − y j (cid:107) + (cid:107) y j − T ∗ diag [Λ( x j )] T (cid:107)≤ (1 + c N ) (cid:107) x j − y j (cid:107) ≤ (1 + c N ) δ. (4.3)If we set V := W T and ε = (1 + c N ) δ , we will have that by 4.2 and 4.3 the inner C ∗ -automorphism Ψ := Ad[ V ∗ ] satisfies the constraints in the statement of thislemma, and we are done. (cid:3) OCAL MATRIX HOMOTOPIES 13
Remark 4.2.
The C ∗ -automorphism Ψ from L.4.1 is called an isospectral approx-imant for the two N -tuples x , . . . , x N and y , . . . , y N . If Ψ := Ad[ W ∗ ] for some W ∈ U ( n ) , then we will have that its inverse Ψ † will be given by the expression Ψ † = Ad[ W ] . Remark 4.3.
The constant c N in the proof of L.4.1 depends only on the number N of matrices in each family. It does not depend on the matrix size. Local piecewise analytic connectivity.
In this section we will present somepiecewise analytic local connectivity results in matrix representations of the form C ε ( T ) → M n ← C ( T N ) and C ε ( J × T ) → M n ← C ( J N ). Theorem 4.1 (Local normal toral connectivity) . Given ε > and any n ∈ Z + ,there is δ > such that, for any N normal contractions x , . . . , x N and y , . . . ,y N in M n which satisfy the relations (cid:26) [ x j , x k ] = [ y j , y k ] = 0 , ≤ j, k ≤ N, (cid:107) x j − y j (cid:107) ≤ δ, ≤ j ≤ N, there exist N toral matrix links X , . . . , X N in M n , which solve the problems x j (cid:32) y j , ≤ j ≤ N, and satisfy the constraints (cid:8) (cid:107) X jt ( x j ) − y j (cid:107) ≤ ε, for each ≤ j, k ≤ N and each t ∈ I . Moreover, (cid:96) (cid:107)·(cid:107) ( X jt ( x j )) ≤ ε, ≤ j ≤ N .Proof. By L.2.1, L.2.2 and L.4.1 we will have that given ε >
0, there are 0 < δ ≤ ν ≤ ε/ W ∗ ] (with W ∈ U ( n )) for x , . . . , x N and y , . . . , y N such that, max {(cid:107) x j − Ψ( x j ) (cid:107) , (cid:107) y j − Ψ( x j ) (cid:107)} ≤ ν and [Ψ( x j ) , y j ] = 0for each 1 ≤ j ≤ N , we will also have that there is a unitary path W ∈ C ( I , M n )which is defined by the expression W t := e − itH W for each t ∈ I , where H W ∈ M n isa hermitian matrix such that e iH W = W and (cid:107) [ H W , x j ] (cid:107) ≤ ε/ ≤ j ≤ N ,and that is defined by H W := h ( W ), for some function h : Ω αd,s → [ − , σ ( W ) ⊂ Ω αd,s := { e i ( πt + α ) | − s < t < − s } ⊂ T , with s, α ∈ R chosenin such a way that T \ Ω αd,s contains an arc of length d (with d ≥ π/n ). Moreover,we can choose δ and ν in such a way that the path W satisfies the inequalities (cid:107) [ W t , Ψ( x j )] (cid:107) ≤ ε/ t ∈ [0 ,
1] and each 1 ≤ j ≤ N .It can be seen that the paths ˘ X jt := Ad[ W t ]( x j ) will solve the problem x j (cid:32) ε/ Ψ( x j ) for each 1 ≤ j ≤ N . Let us set ¯ X jt := (1 − t )Ψ( x j ) + ty j , we can nowconstruct N toroidal matrix links of the form X j := ˘ X j (cid:126) ¯ X j which solve theproblems x j (cid:32) y j , locally preserve normality and commutativity and satisfy the (cid:107) · (cid:107) -distance constraints (cid:107) X jt − y j (cid:107) ≤ (cid:107) X jt − Ψ( x j ) (cid:107) + (cid:107) y j − Ψ( x j ) (cid:107)≤ ε ν ≤ ε ε ε, together with the (cid:107) · (cid:107) -length constraints (cid:96) (cid:107)·(cid:107) ( X jt ) ≤ (cid:96) (cid:107)·(cid:107) ( ˘ X jt ) + (cid:107) Ψ( x j ) − y j (cid:107) (4.4) = (cid:90) I (cid:107) ∂ t Ad[ W t ]( x j ) (cid:107) dt + (cid:107) Ψ( x j ) − y j (cid:107) (4.5) = (cid:107) [ H W , Ψ( x j )] (cid:107) + (cid:107) Ψ( x j ) − y j (cid:107) (4.6) ≤ ε ν ≤ ε, (4.7)which hold whenever (cid:107) x j − y j (cid:107) ≤ δ , 1 ≤ j ≤ N , and we are done. (cid:3) Remark 4.4.
It can be noticed that the solvent matrix links X , . . . , X N whoseexistence is stated in T.4.1 are factored in the form X j = ˘ X j (cid:126) ¯ X j , we call ˘ X j and ¯ X j the curved and flat factors of X j respectively. We will derive now, some corollaries of the proof of T.4.1.
Corollary 4.1 (Local hermitian toral connectivity) . Given ε > and any integer n ≥ , there is δ > such that, for any N hermitian contractions x , . . . , x N and y , . . . , y N in M n which satisfy the relations (cid:26) [ x j , x k ] = [ y j , y k ] = 0 , ≤ j, k ≤ N, (cid:107) x j − y j (cid:107) ≤ δ, ≤ j ≤ N, there exist N toral matrix links X , . . . , X N in M n , which solve the problems x j (cid:32) y j , ≤ j ≤ N, and satisfy the constraints (cid:26) X jt ( x j ) = ( X jt ( x j )) ∗ , (cid:107) X jt ( x j ) − y j (cid:107) ≤ ε, for each ≤ j, k ≤ N and each t ∈ I . Moreover, (cid:96) (cid:107)·(cid:107) ( X jt ( x j )) ≤ ε, ≤ j ≤ N .Proof. Since for any α ∈ R , any pair of hermitian matrices x, y ∈ H ( n ) and anypartial unitary z ∈ PU ( n ), we have that x + α ( y − x ) and zxz ∗ are also in H ( n ),the result follows as a consequence of L.4.1 and T.4.1. (cid:3) Corollary 4.2 (Local unitary toral connectivity) . Given any ε ≥ and any in-teger n ≥ , there is δ ≥ such that given any N unitary matrices U , . . . , U N , V , . . . , V N in M n which satisfy the relations (cid:26) [ U j , U k ] = [ V j , V k ] = 0 , (cid:107) U k − V k (cid:107) ≤ δ, for each ≤ j, k ≤ N , there are toral matrix links u , . . . , u N in M n which solvethe interpolation problems U k (cid:32) V k , ≤ k ≤ N, and also satisfy the relations (cid:26) ( u jt ) ∗ u jt = u jt ( u jt ) ∗ = n , (cid:107) u jt − V j (cid:107) ≤ ε, for each t ∈ I and each ≤ j, k ≤ N . Moreover, (cid:96) (cid:107)·(cid:107) ( u jt ) ≤ ε , ≤ j ≤ N . OCAL MATRIX HOMOTOPIES 15
Proof.
Since for any C ∗ -automorphisms Ψ we have that Ψ( U ( n )) ⊆ U ( n ), andsince any two commuting unitaries U and V can connected by a flat unitary path¯ U t := U e t ln( U ∗ V ) , for 0 ≤ t ≤
0. We will have that the result can be derived usinga similar argument to the one implemented in the proof of T.4.1. (cid:3)
Lifted local piecewise analytic connectivity.
Let us denote by κ the matrixcompression M n → M n defined by the mapping κ : M n → M n , (cid:18) x x x x (cid:19) (cid:55)→ x . Let us write ı : M n → M n to denote the C ∗ -homomorphism defined by theexpression ı ( x ) := x ⊕ x = ⊗ x . Definition 4.2 (Standard dilations) . Given a C ∗ -automorphism Ψ := Ad[ W ] (with W ∈ U ( n ) ) in M n , we will denote by Ψ [ s ] the C ∗ -automorphism in M n defined bythe expression Ψ [ s ] := Ad[ ⊗ W ] = Ad[ W ⊕ W ] . We call Ψ [ s ] a standard dilationof Ψ . Definition 4.3 ( Z / . Given a C ∗ -automorphism Ψ := Ad[ W ] (with W ∈ U ( n ) ) in M n , we will denote by Ψ [2] the C ∗ -automorphism in M n defined bythe expression Ψ [2] := Ad[(Σ ⊗ n )( W ∗ ⊕ W )] . We call Ψ [2] a Z / -dilation of Ψ . Remark 4.5.
It can be seen that κ ( ı ( x )) = x for any x ∈ M n , it can also be seenthat κ (Ψ [2] ( ı ( x ))) = κ (Ψ [ s ] ( ı ( x ))) . Theorem 4.2 (Lifted local toral connectivity) . Given ε > , there is δ > suchthat, for any N normal contractions x , . . . , x N and y , . . . , y N in M n which satisfythe relations (cid:26) [ x j , x k ] = [ y j , y k ] = 0 , ≤ j, k ≤ N, (cid:107) x j − y j (cid:107) ≤ δ, ≤ j ≤ N, there is a C ∗ -homomorphism Φ : M n → M n and N toral matrix links X , . . . , X N in C ( I , M n ) , which solve the problems Φ( x j ) (cid:32) y j ⊕ y j , ≤ j ≤ N, and satisfy the constraints κ (Φ( x j )) = x j , (cid:107) Φ( x j ) − x j ⊕ x j (cid:107) ≤ ε, (cid:107) X jt − y j ⊕ y j (cid:107) ≤ ε, for each ≤ j, k ≤ N and each t ∈ I . Moreover, (cid:96) (cid:107)·(cid:107) ( X jt ) ≤ ε, ≤ j ≤ N .Proof. By L.4.1 we will have that given ε >
0, there are 0 < δ ≤ ν = ε π andan isospectral approximant Ψ := Ad[ W ∗ ] (with W ∈ U ( n )) for x , . . . , x N and y , . . . , y N such that, max {(cid:107) x j − Ψ( x j ) (cid:107) , (cid:107) y j − Ψ( x j ) (cid:107)} ≤ ν . By setting Φ :=(Ψ † ) [2] ◦ ı ◦ Ψ, by D.4.3, D.4.2 and R.4.5 it can be seen that Φ : M n → M n is a C ∗ -homomorphism such that (cid:107) Φ( x j ) − ı ( x j ) (cid:107) = (cid:107) Φ( x j ) − x j ⊕ x j (cid:107) ≤ ε , for each1 ≤ j ≤ N .Since (Ψ † ) [2] := Ad[ ˆ W s ] with ˆ W s := (Σ ⊗ n )( W ∗ ⊕ W ) and since ˆ W s ∈ U (2 n ) ∩ H (2 n ), we will have that ˆ W s can be represented as ˆ W s = e i π ( ˆ W s − n ) for any n ≥
1. If we set ˜ X j := Ψ [ s ] ( ı ( x j )), 1 ≤ j ≤ N , we also have that there is a unitary path {W t } t ∈ I ⊂ M n with W t := e i π (1 − t )2 ( ˆ W s − n ) , which satisfies the conditions W = ˆ W s , W = n , together with the normed estimates (cid:107)W t ˜ X j − ˜ X j W t (cid:107) = | cos( πt/ |(cid:107) ˆ W s ˜ X j − ˜ X j ˆ W s (cid:107)≤ (cid:107) ˆ W s ˜ X j − ˜ X j ˆ W s (cid:107) ≤ ν, for each 1 ≤ j ≤ N and each 0 ≤ t ≤
1. Moreover, for each 1 ≤ j ≤ N we havethat the paths ˘ X jt := Ad[ W t ]( ˜ X j ) satisfy the normed estimates (cid:96) (cid:107)·(cid:107) ( ˘ X jt ) = (cid:90) I (cid:107) ∂ t Ad[ W t ]( ˜ X j )) (cid:107) dt, = π (cid:107) ˆ W s ˜ X j − ˜ X j ˆ W s (cid:107) ≤ ν. For each 1 ≤ j ≤ N , we can now use the flat paths ¯ X jt := (1 − t ) ˜ X j + tı ( y j )together with the previously described curved paths ˘ X j to construct the solventtoral matrix links X , . . . , X N ∈ C ([0 , , M n ) we are looking for, and which canbe defined by X j := ˘ X j (cid:126) ¯ X j for each 1 ≤ j ≤ N , and we are done. (cid:3) Remark 4.6.
It can be seen that by using the technique implemented in the proofof T.4.2 one can obtain lifted versions of C.4.2 and C.4.1.
Remark 4.7.
As a consequence of T.4.2 we can derive simple detection methodsto identify families of pairwise commuting matrices in M n that can be connecteduniformly via piecewise analytic toral matrix links. The existence of these detectionmethods raises some interesting questions for further studies. Remark 4.8.
We can interpret T.4.2 as an existence theorem of solutions to liftedconnectivity problems defined on matrix representations of the form C ∗ ε (cid:104) Z / × Z (cid:105) (cid:47) (cid:47) C ∗ ( ˆ U s , ˆ V ) (cid:47) (cid:47) M n (cid:15) (cid:15) C ∗ (cid:104) F (cid:105) (cid:47) (cid:47) (cid:56) (cid:56) C δ ( T ) (cid:47) (cid:47) C ∗ ( U, V ) (cid:79) (cid:79) (cid:47) (cid:47) M n , with ˆ U s = (Σ ⊗ n )( U ∗ ⊕ U ) and ˆ V = V ⊕ V . Some further applications of T.4.2 to approximation of matrix words and normbehavior will be presented in [33].4.2.2.
Matrix Klein Bottles: Local matrix deformations and special symmetries.
Using T.4.2 we can solve all connectivity problems (together with their softenedversions) in M n that can be reduced to connectiviy problems of the form x (cid:32) ε x ∗ in N ( n )( D ), with x ∗ = T xT and T = n . Remark 4.9.
For each ε ∈ [0 , , we can use the previously described symmetriesand D T to interpret (cid:83) x ∈ M n { x (cid:32) ε x ∗ } as matrix analogies of the Klein bottle . By a softened matrix Klein bottle we mean that the symmetries are softened,in particular we can consider the connectiviy problems x (cid:32) ε x ∗ and y (cid:32) ε y ∗ in N ( n )( D ) subject to the normed constraints (cid:107) xy − yx (cid:107) ≤ δ , (cid:107) x ∗ − T xT (cid:107) , (cid:107) xT − T y (cid:107) ≤ δ and T = n . The details regarding to the solvability of these localconnectivity problems will be addressed in future communications. OCAL MATRIX HOMOTOPIES 17 C uniform local connectivity of pairs of unitaries and piecewiseanalytic approximants. The technique presented in this section can be used tosolve local connectivity problems in matrix representations of the form C ε ( T ) → M n ← C ( T ) uniformly via C -unitary paths.Suppose U t and V t are unitary matrices in M n ( C ) for t = 0 and t = 1 and wedefine(4.8) U t = U e t ln( U ∗ U ) and(4.9) V t = V e t ln( V ∗ V ) . For t = 0 or t = 1 the C ∗ -algebra generated by U t and V t is abelian, so select aMASA C t ∼ = C n in each case. Let A ( C , C ) = { X ∈ C ([0 , , M n ( C )) | X (0) ∈ C and X (1) ∈ C } . Lemma 4.2.
The C ∗ -algebra A ( C , C ) has stable rank one.Proof. Starting with X continuous with X ( t ) in C t at the endpoints, we can adjustthis by a small amount, leaving the endpoints in C t , to get X piece-wise linear,with the endpoints of every linear segment having no spectral multiplicity andbeing invertible. Using Kato’s theory of analytic paths, we can get a piece-wisecontinuous unitary U t and piece-wise analytic scalar paths λ n ( t ) so that the newpath Y ≈ X satisfies Y ( t ) = U t λ ( t ) . . . λ n ( t ) U ∗ t . There may be finitely may places where Y ( t ) is not invertible. These places willbe in the interior of the segment so in an open interval where U t is continuous. Asmall deformation of some of the λ j will take the path through invertibles. We havenot moved the endpoints in the second adjustment so the constructed element is in A ( C , C ) and close to X . (cid:3) Lemma 4.3.
The endpoint-restriction map ρ : A ( C , C ) → C ⊕ C induces aninjection on K .Proof. The kernel of ρ is C ([0 , , M n ( C )) which has trivial K -group. So this resultfollows from the exactness of the usual six-term sequence in K -theory. (cid:3) Lemma 4.4.
Given unitaries U and V in A ( C , C ) , with (cid:107) [ U, V ] (cid:107) ν as if D.2.2(so the Bott index makes sense), Bott(
U, V ) is the trivial element of K ( A ( C , C )) .Proof. By the previous lemma, we need only calculate Bott( ρ ( U ) , ρ ( V )). Theseunitaries are in a commutative C ∗ -algebra so they have trivial Bott index. (cid:3) Theorem 4.3.
Given (cid:15) > , there exists δ > so that for all n , given unitarymatrices U , U , V , V in M n ( C ) with U V = V U , U V = V U , (cid:107) U − U (cid:107) ≤ δ and (cid:107) V − V (cid:107) ≤ δ , then there exists continuous paths U t and V t between the givenpairs of unitaries with each U t and V t unitary, and with U t V t = V t U t , (cid:107) U t − U (cid:107) ≤ (cid:15) and (cid:107) V t − V (cid:107) ≤ (cid:15) for all t . Proof.
The paths U t and V t defined in equations 4.8 and 4.9 will be almost com-muting unitary elements of A ( C , C ). By Lemma 4.2 we may apply [9, Theorem8.1.1] regarding approximating in A ( C , C ) by commuting unitaries. Lemma 4.4tells us there is no invariant to worry about, so we can find A t and B t close of U t and V t that are commuting continuous paths of unitaries with A t and B t in C t for t = 0 ,
1. The unitary elements in the commutative C t are locally connected, so wecan find a short path from U and V to A and B , and likewise at the other end.Concatenating, we get a paths of commuting unitary matrices from U and V to U and V so that at every point we are close to some pair ( U t , V t ). These then areall close to U and V . (cid:3) By combining T.4.2, C.4.2 and T.4.3 it can be seen that.
Remark 4.10 (Piecewise analytic approximants of C interpolants) . Given (cid:15) > ,there exists δ > so that for all n , given unitary matrices U , U , V , V in M n ( C ) with U V = V U , U V = V U , (cid:107) U − U (cid:107) ≤ δ and (cid:107) V − V (cid:107) ≤ δ , thereexist continuous (interpolants) paths U t and V t in M n which solve the problems U ⊕ U (cid:32) U ⊕ U and V ⊕ V (cid:32) V ⊕ V with each U t and V t unitary, and with U t V t = V t U t , (cid:107) U t − U ⊕ U (cid:107) ≤ (cid:15) and (cid:107) V t − V ⊕ V (cid:107) ≤ (cid:15) for all t . There are alsoa C ∗ -homomorphism Ψ : M n → M n such that max {(cid:107) Ψ( U ) − U ⊕ U (cid:107) , (cid:107) Ψ( U ) − U ⊕ U (cid:107) , (cid:107) Ψ( V ) − V ⊕ V (cid:107) , (cid:107) Ψ( V ) − V ⊕ V (cid:107)} ≤ (cid:15), and two piecewise analytic unitary pairwise commuting paths ˆ U , ˆ V ∈ C ([0 , , M n ) which solve the problems Ψ( U ) (cid:32) U ⊕ U , Ψ( V ) (cid:32) V ⊕ V with max {(cid:107) ˆ U t − U t (cid:107) , (cid:107) ˆ V t − V t (cid:107)} ≤ (cid:15) for each ≤ t ≤ . Moreover, (cid:96) (cid:107)·(cid:107) ( ˆ U t ) ≤ (cid:15) and (cid:96) (cid:107)·(cid:107) ( ˆ V t ) ≤ (cid:15) . Jointly compressible matrix sets.
Given 0 < δ ≤ ε , we can now consideran alternative approach to the local connectivity problem involving two N -sets ofpairwise commuting normal matrix contractions X , . . . , X N and Y , . . . , Y N suchthat (cid:107) X j − Y j (cid:107) ≤ δ for each 1 ≤ j ≤ N . The approach that we will consider inthis section consists of considering the existence of a normal contraction ˆ X suchthat X , . . . , X N ∈ C ∗ ( ˆ X ), and which also satisfies the constraint (cid:107) ˆ X − X j (cid:107) ≤ ε for some 1 ≤ j ≤ N . A matrix ˆ X which satisfies the previous conditions will becalled a nearby generator for X , . . . , X N , it can be seen that for any δ ≤ ν ≤ ε one can find a flat analytic path ¯ X ∈ C ([0 , , M ∞ ) that performs the deformation X j (cid:32) ν ˆ X , where ˆ X is a nearby generator for X , . . . , X N .Given any joint isospectral approximant Ψ with respect to the families normalcontractions described in the previous paragraph, along the lines of the programthat we have used to derive the connectivity results T.4.1 and T.4.2, we can useL.4.1 to find a C ∗ -automorphism which solves the extension problem described bythe diagram,(4.10) C ∗ ( ˆ X ) ˆΨ (cid:15) (cid:15) C ∗ ( X , . . . , X N ) (cid:40) (cid:8) (cid:54) (cid:54) Ψ (cid:47) (cid:47) C ∗ ( Y , . . . , Y N ) OCAL MATRIX HOMOTOPIES 19 and satisfies the relations Ψ( X j ) = ˆΨ( X j ) for each 1 ≤ j ≤ N together with thenormed constraintsmax {(cid:107) ˆΨ( ˆ X ) − ˆ X (cid:107) , max j {(cid:107) ˆΨ( X j ) − X j (cid:107) , (cid:107) ˆΨ( X j ) − Y j (cid:107)}} ≤ ε. We refer to the C ∗ -automorphism ˆΨ in 4.10 as a compression of Ψ or a com-pressive joint isospectral approximant ( CJIA ) for the N -sets of normal con-tractions. Let us now consider a special type of inner C ∗ -automorphisms that canbe described as follows. Definition 4.4 (Uniformly compressible JIA) . Given < δ ≤ ε and two N -sets ofpairwise commuting normal contractions X , . . . , X N and Y , . . . , Y N in M ∞ suchthat (cid:107) X j − Y j (cid:107) ≤ δ , ≤ j ≤ N , a joint isospectral approximant Ψ of the N -sets is said to be uniformly compressible if there are a nearby generator ˆ X for X , . . . , X N , a compression ˆΨ := Ad[ W ] (with W ∈ U ( M ∞ ) ) of Ψ and a unitary ˆ W ∈ ˆΨ( C ∗ ( ˆ X )) such that (cid:107) W − ˆ W (cid:107) ≤ ε . We refer to the N normal contractions X , . . . , X N and Y , . . . , Y N for which there exists a uniformly compressible JIA( UCJIA ) as uniformly jointly compressible (
UJC ). Lemma 4.5 (Local connectivity of
UJC matrix sets) . Given ε > , there is δ > ,such that for any two N -sets of UJC pairwise commuting normal contractions X , . . . , X N and Y , . . . , Y N in M ∞ such that (cid:107) X j − Y j (cid:107) ≤ δ for each ≤ j ≤ N ,we will have that there are N toral matrix links X , . . . , X N ∈ C ([0 , , M ∞ ) thatsolve the interpolation problem X j (cid:32) ε Y j , for each ≤ j ≤ N .Proof. Since the N -sets of pairwise commuting normal contractions X , . . . , X N and Y , . . . , Y N are UJC , we have that given 0 < δ ≤ ν ≤ ε/ <
1, there area normal contraction ˆ X ∈ M ∞ which commutes with each X j together with a UCJIA ˆΨ = Ad[ W ] for some W ∈ U ( M ∞ ) and a unitary ˆ W ∈ ˆΨ( C ∗ ( ˆ X )) suchthat(4.11) (cid:107) − ˆ W ∗ W (cid:107) = (cid:107) W − ˆ W (cid:107) ≤ ν < . Let us set Z := ˆ W ∗ W , as a consequence of the inequality 4.11 we will have thatthere is a hermitian matrix − ≤ H Z ≤ in M ∞ such that e πiH Z = Z . Byusing 4.11 again, it can be seen that we can now use the curved paths ˘ X j :=Ad[ e − πitH Z ]( X j ) to solve the problems X j (cid:32) ε/ ˆΨ( X j ), and then we can solvethe problems ˆΨ( X j ) (cid:32) ν Y j using the flat paths ¯ X j := (1 − t ) ˆΨ( X j ) + tY j . Wecan construct the solvent toral matrix links by setting X j := ˘ X j (cid:126) ¯ X j for each1 ≤ j ≤ N . This completes the proof. (cid:3) Hints and Future Directions
The detection matrix representations of universal C ∗ -algebras that can be con-nected uniformly via piecewise analytic paths induces interesting problems whichare topological/K-theoretical and computational in nature. Motivated by the C -connectivity technique, we consider that the use of T.4.1, C.4.2 and T.4.2 and L.4.5to study local matrix connectivity in C ∗ -representations of the form C ( T N ) (cid:41) (cid:41) (cid:47) (cid:47) M n M n (cid:111) (cid:111) C ε ( T ) (cid:111) (cid:111) (cid:105) (cid:105) will present interesting challenges and questions that will be the subject of futurestudy. In particular we are interested in the application of T.4.2, C.4.2 and L.4.5 to the study of the question. Is C ∗ (cid:104) F × F (cid:105) RFD? (This is equivalent to
Connes’sembedding problem .)A better understanding of the geometric and approximate combinatorial natureof toroidal matrix links would provide a mutually benefitial interaction betweenmatrix flows in the sense of Brockett [4] and Chu [8], topologically controlled linearalgebra in the sense of Freedman and Press [14] and matrix geometric deformationsin the sense of Loring [25]. This also may provide some novel generic numericalmethods to study and compute normal matrix compressions, sparse representationsand dimensionality reduction of large scale matrices. Using a similar approach weplan to use T.4.2 and L.4.5 to answer some questions in topologically controlledlinear algebra in the sense of [14], raised by M. H. Freedman.The construction and generalization of detection methods like the ones men-tioned in the remark R.4.7 of theorem T.4.2 together with their implications oninverse spectral problems, will be the subject of future studies. In particular wewill use toral matrix links to study the local deformation properties of matrix rep-resentations of the form C ε ( T ) (cid:111) α Z / → M n (where α denotes the standard flip)via softened matrix Klein bottles . These problems are related to spectral decom-position problems with spectral symmetry in quantum theory and to deformationtheory for C ∗ -algebras in the sense of Loring [25]. We will also use toroidal matrixlinks to study the local connectivity of some Soft group C ∗ -algebras in the sense ofFarsi [13].Some generalizations of T.4.2 and particular applications of L.4.5 to the studyof matrix equations on words will also be the subject of future study. In particular,the combination of toroidal matrix links with some matrix lifting techniques alongthe same lines of the proof of T.4.2 combined with L.4.5, seem also promising onthe solvability of some conjectures studied numerically on [28].6. Acknowledgement
Both authors are very grateful with the Erwin Schr¨odinger Institute for Mathe-matical Physics of the University of Vienna, for the outstanding hospitality duringour visit to participate in the research program on Topological phases of quan-tum matter in August of 2014. Much of the research reported in this documentwas carried out while we were visiting the Institute. The second author wants tothank Moody T. Chu for his warm hospitality during his visit to the Department ofMathematics at North Carolina State, for precise comments, challenging questions,and for sharing interesting conjectures and problems with him. He also wants tothank Alexandru Chirvasitu for several interesting questions and comments thathave been very helpful for the preparation of this document.This work was partially supported by a grant from the Simons Foundation(208723 to Loring).
References [1]
R. Bhatia.
Matrix Analysis.
Gaduate Texts in Mathematics 169. Springer-Verlag. 1997.[2]
R. Bhatia.
Positive Definite Matrices.
Princeton University Press. 2007.[3]
O. Bratteli, G. A. Elliot, D. E. Evans and A. Kishimoto.
Homotopy of a Pair of Approx-imately Commuting Unitaries in a Simple C*-Algebra.
J. Funct. Anal. 160, 466-523 (1998)Article No. FU983261[4]
R. W. Brockett.
Least Squares Matching Problems.
Linear Algebra Appl. 122/123/124: 761-777 (1989).
OCAL MATRIX HOMOTOPIES 21 [5]
M.-D. Choi and E. G. Effros.
The completely positive lifting problem for C*-algebras.
Ann.of Math., 104 (1976), 585-609[6]
M.-D. Choi.
The Full C*-Algebra of the Free Group on Two Generators.
Pacific J. Math.,Vol. 87, No. 1, 1980.[7]
M. T. Chu.
A Simple Application of the Homotopy Method to Symmetric Eigenvalue Prob-lems.
Linear Algebra Appl. 59:85-90 (1984).[8]
M. T. Chu.
Linear Algebra Algorithms as Dynamical Systems.
Acta Numerica (2008), pp.001-086. 2008.[9]
S. Eilers, T. A. Loring and G. K. Pedersen.
Stability of anticommutation relations: anapplication of noncommutative CW complexes.
J. Reine Angew. Math., 499:101-143, 1998.[10]
S. Eilers and R. Exel.
Finite Dimensional Representations of the Soft Torus.
Proc. Amer.Math. Soc. Vol. 130, No. 3 (Mar., 2002), pp. 727-731.[11]
L. Elsner.
Perturbation Theorems for the Joint Spectrum of Commuting Matrices: A Con-servative Approach.
Linear Algebra Appl. 208/209:83-95 (1994)[12]
R. Exel and T. A. Loring.
Invariants of Almost Commuting Unitaries.
J. Funct. Anal. 95,364-376 (1991).[13]
C. Farsi.
Soft Non-commutative Toral C*-Algebras.
J. Funct. Anal. 151, 35-49 (1997) ArticleNo. FU97313[14]
M. H. Freedman and W. H. Press.
Truncation of Wavelet Matrices: Edge Effects and theReduction of Topological Control
Linear Algebra Appl. 2:34:1-19 (1996)[15]
M. B. Hastings and T. A. Loring.
Topological insulators and C*-algebras: Theory andnumerical practice.
Ann. Physics, 326(7):1699–1759, 2011.[16]
R. C. Kirby.
Stable homeomorphisms and the annulus conjecture.
Ann. of Math., SecondSeries, Vol. 89, No. 3 (May, 1969), pp. 575-582[17]
H. Lin.
Almost commuting selfadjoint matrices and applications.
In Fields Inst. Commun.Amer. Math. Soc., volume 13, pages 193-233. Providence, RI, 1997.[18]
H. Lin.
Approximate Homotopy of Homomorphisms from C ( X ) into a Simple C ∗ -algebra. Mem. Amer. Math. Soc., Volume 205 Number 963, 2010.[19]
H. Lin.
An introduction to the classification of amenable C ∗ -algebras. World Scientific, RiverEdge, NJ, ISBN: 981-02-4680-3, pp 1–320. 2001.[20]
H. Lin.
Classification of simple C*-algebras and higher dimensional noncommutative tori.
Ann. of Math., 157 (2003), 521–544[21]
H. Lin.
Homotopy of unitaries in simple C*-algebras with tracial rank one.
J. Funct. Anal.Volume 258, Issue 6, 15 March 2010, Pages 1822–1882[22]
H. Lin
Approximately diagonalizing matrices over C(Y) . Proc. Natl. Acad. Sci. U S A. 2012Feb 21;109(8):2842-7.[23]
T. Loring.
The Torus and Noncommutative Topology.
Ph.D. thesis, University of California,Berkeley, 1986.[24]
T. A. Loring.
K-theory and asymptotically commuting matrices.
Canad. J. Math. 40 (1988),197-216.[25]
T. A. Loring.
Deformations of nonorientable surfaces as torsion E-theory elements.
C. R.Acad. Sci. Paris, t. 316, S´erie I, p. 341-346, 1993.[26]
T. A. Loring.
Lifting solutions to perturbing problems in C*-algebras.
Volume 8 of FieldsInst. Mon. Amer. Math. Soc., Providence, RI, 1997.[27]
T. A. Loring and T. Shulman.
Noncommutative Semialgebraic Sets and Associated LiftingProblems.
Trans. Amer. Math. Soc., 364:721–744, 2012.[28]
T. A. Loring and F. Vides.
Estimating Norms of Commutators.
Experimental MathematicsVol. 24, Iss. 1, 2015.[29]
A. McIntosh, A. Pryde and W. Ricker.
Systems of Operator Equations and Perturbationof Spectral Subspaces of Commuting Operators.
Michigan Math. J. Volume 35, Issue 1 (1988),43-65.[30]
A. J. Pryde. inequalities for the Joint Spectrum of Simultaneously Triangularizable Ma-trices.
Proc. Centre Math. Appl., Mathematical Sciences Institute, The Australian NationalUniversity (1992), 196-207.[31]
M. Rørdam, F. Larsen and N. J. Laustsen.
An Introduction to K-Theory for C*-Algebras.
London Math. Soc., Student Texts 49. 2000.[32]
F. Vides.
Toroidal Matrix Links: Local Matrix Homotopies and Soft Tori.
Ph.D. thesis, TheUniversity of New Mexico, Albuquerque, 2016. [33]
F. Vides.
Local Deformation of Matrix Words.
In preparation.
Department of Mathematics and StatisticsThe University of New Mexico, Albuquerque, NM 87131, USA.
Current address , F. Vides:
School of Mathematics and Computer ScienceNational Autonomous University of Honduras, Ciudad Universitaria, 2do Piso,Edificio F1, Honduras, C.A.
E-mail address , T. A. Loring: [email protected]
E-mail address , F. Vides:, F. Vides: