Almost commuting unitary matrices Adam P. W. Sørensen University of Wollongong

May, 2013

MODEL / ANALYSE / FORMULATE / ILLUMINATE CONNECT:IMIA

Almost Commuting Matrices Problem Given any two square matrices A, B satisfying 1. kAB − BAk is small and 2. A, B both belong to some class of matrices C, can we find two matrices X , Y such that 1. X , Y commute, 2. kX − Ak, kY − Bk are small, and, 3. X , Y belong to C?

Theorem (Lin) If C is the class of self-adjoint matrices, then the problem has a positive solution.

Adam P. W. Sørensen

Almost commuting unitaries 2 / 5

Almost Commuting Matrices Problem For every ε > 0 can we find a δ > 0 such that: Given any two square matrices A, B satisfying 1. kAB − BAk < δ and 2. A, B both belong to some class of matrices C, we can find two matrices X , Y such that 1. X , Y commute, 2. kX − Ak, kY − Bk < ε, and, 3. X , Y belong to C.

Theorem (Lin) If C is the class of self-adjoint matrices, then the problem has a positive solution.

Adam P. W. Sørensen

Almost commuting unitaries 2 / 5

Almost Commuting Matrices Problem For every ε > 0 can we find a δ > 0 such that: Given any two square matrices A, B satisfying 1. kAB − BAk < δ and 2. A, B both belong to some class of matrices C, we can find two matrices X , Y such that 1. X , Y commute, 2. kX − Ak, kY − Bk < ε, and, 3. X , Y belong to C.

Theorem (Lin) If C is the class of self-adjoint matrices, then the problem has a positive solution.

Adam P. W. Sørensen

Almost commuting unitaries 2 / 5

Almost Commuting Unitaries Example (Voiculescu, Exel-Loring) For each n let ωn = exp  ωn    Ωn =   

2πi n




, define 

ωn2

ωn3 ..

. ωnn

   ,  

 0 1   Sn =   

0 1

0 ..

. 1

Then 1. kSn Ωn − Ωn Sn k → 0 as n → 0, 2. For any two commuting matrices X , Y we have p max {kSn − X k, kΩn − Y k} > 2 − |ωn − 1| − 1.

Adam P. W. Sørensen

Almost commuting unitaries 3 / 5

 1    .   0

Almost Commuting Unitaries Example (Voiculescu, Exel-Loring) For each n let ωn = exp  ωn    Ωn =   

2πi n




, define 

ωn2

ωn3 ..

. ωnn

   ,  

 0 1   Sn =   

0 1

0 ..

. 1

Then 1. kSn Ωn − Ωn Sn k → 0 as n → 0, 2. For any two commuting matrices X , Y we have p max {kSn − X k, kΩn − Y k} > 2 − |ωn − 1| − 1.

Adam P. W. Sørensen

Almost commuting unitaries 3 / 5

 1    .   0

Almost Commuting Unitaries Example (Voiculescu, Exel-Loring) For each n let ωn = exp  ωn    Ωn =   

2πi n




, define 

ωn2

ωn3 ..

. ωnn

   ,  

 0 1   Sn =   

0 1

0 ..

. 1

Then 1. kSn Ωn − Ωn Sn k → 0 as n → 0, 2. For any two commuting matrices X , Y we have p max {kSn − X k, kΩn − Y k} > 2 − |ωn − 1| − 1.

Adam P. W. Sørensen

Almost commuting unitaries 3 / 5

 1    .   0

Additional Symmetry The Dual of a Matrix 

A C

B D

]

DT −C T

 =

−B T AT



Question What happens if our matrices are required to be symmetric or self-]? 1. Lin’s theorem works: Close to almost commuting self-adjoint symmetric (or self-]) matrices we can find commuting self-adjoint (or self-]) matrices. (Loring-S) 2. The invariant that prevent almost commuting unitaries from being close to exactly commuting once, vanish for both symmetric and self-] unitaries. (Hastings-Loring) 3. There is an obstruction preventing almost commuting self-] unitaries from being close to exactly commuting self-] unitaries. (Hastings-Loring) Adam P. W. Sørensen

Almost commuting unitaries 4 / 5

Additional Symmetry The Dual of a Matrix 

A C

B D

]

DT −C T

 =

−B T AT



Question What happens if our matrices are required to be symmetric or self-]? 1. Lin’s theorem works: Close to almost commuting self-adjoint symmetric (or self-]) matrices we can find commuting self-adjoint (or self-]) matrices. (Loring-S) 2. The invariant that prevent almost commuting unitaries from being close to exactly commuting once, vanish for both symmetric and self-] unitaries. (Hastings-Loring) 3. There is an obstruction preventing almost commuting self-] unitaries from being close to exactly commuting self-] unitaries. (Hastings-Loring) Adam P. W. Sørensen

Almost commuting unitaries 4 / 5

Additional Symmetry The Dual of a Matrix 

A C

B D

]

DT −C T

 =

−B T AT



Question What happens if our matrices are required to be symmetric or self-]? 1. Lin’s theorem works: Close to almost commuting self-adjoint symmetric (or self-]) matrices we can find commuting self-adjoint (or self-]) matrices. (Loring-S) 2. The invariant that prevent almost commuting unitaries from being close to exactly commuting once, vanish for both symmetric and self-] unitaries. (Hastings-Loring) 3. There is an obstruction preventing almost commuting self-] unitaries from being close to exactly commuting self-] unitaries. (Hastings-Loring) Adam P. W. Sørensen

Almost commuting unitaries 4 / 5

Additional Symmetry The Dual of a Matrix 

A C

B D

]

DT −C T

 =

−B T AT



Question What happens if our matrices are required to be symmetric or self-]? 1. Lin’s theorem works: Close to almost commuting self-adjoint symmetric (or self-]) matrices we can find commuting self-adjoint (or self-]) matrices. (Loring-S) 2. The invariant that prevent almost commuting unitaries from being close to exactly commuting once, vanish for both symmetric and self-] unitaries. (Hastings-Loring) 3. There is an obstruction preventing almost commuting self-] unitaries from being close to exactly commuting self-] unitaries. (Hastings-Loring) Adam P. W. Sørensen

Almost commuting unitaries 4 / 5

Additional Symmetry The Dual of a Matrix 

A C

B D

]

DT −C T

 =

−B T AT



Question What happens if our matrices are required to be symmetric or self-]? 1. Lin’s theorem works: Close to almost commuting self-adjoint symmetric (or self-]) matrices we can find commuting self-adjoint (or self-]) matrices. (Loring-S) 2. The invariant that prevent almost commuting unitaries from being close to exactly commuting once, vanish for both symmetric and self-] unitaries. (Hastings-Loring) 3. There is an obstruction preventing almost commuting self-] unitaries from being close to exactly commuting self-] unitaries. (Hastings-Loring) Adam P. W. Sørensen

Almost commuting unitaries 4 / 5

Additional Symmetry Contd. Theorem (Loring-S) For every ε > 0 there exists a δ > 0 such that for any two unitary matrices U, V with 1. U T = U, V T = V , and, 2. kUV − VUk ≤ δ, there exist unitary matrices U 0 , V 0 with 1. U 0T = U 0 , V 0T = V 0 , 2. kU − U 0 k, kV − V 0 k ≤ ε, and 3. U 0 V 0 = V 0 U 0 .

Theorem (Loring-S) The only obstruction preventing almost commuting self-] unitaries from being close to commuting self-] unitaries is the one found by Hastings and Loring. Adam P. W. Sørensen

Almost commuting unitaries 5 / 5

Additional Symmetry Contd. Theorem (Loring-S) For every ε > 0 there exists a δ > 0 such that for any two unitary matrices U, V with 1. U T = U, V T = V , and, 2. kUV − VUk ≤ δ, there exist unitary matrices U 0 , V 0 with 1. U 0T = U 0 , V 0T = V 0 , 2. kU − U 0 k, kV − V 0 k ≤ ε, and 3. U 0 V 0 = V 0 U 0 .

Theorem (Loring-S) The only obstruction preventing almost commuting self-] unitaries from being close to commuting self-] unitaries is the one found by Hastings and Loring. Adam P. W. Sørensen

Almost commuting unitaries 5 / 5