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application of c*algebras to PDEs
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I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. Does anyone know concrete connections between C*algebras and the theory of PDEs? Best regards.
functional-analysis pde operator-algebras c-star-algebras banach-algebras
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
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add a comment |
$begingroup$
I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. Does anyone know concrete connections between C*algebras and the theory of PDEs? Best regards.
functional-analysis pde operator-algebras c-star-algebras banach-algebras
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
$endgroup$
add a comment |
$begingroup$
I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. Does anyone know concrete connections between C*algebras and the theory of PDEs? Best regards.
functional-analysis pde operator-algebras c-star-algebras banach-algebras
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
$endgroup$
I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. Does anyone know concrete connections between C*algebras and the theory of PDEs? Best regards.
functional-analysis pde operator-algebras c-star-algebras banach-algebras
functional-analysis pde operator-algebras c-star-algebras banach-algebras
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
edited Dec 6 '14 at 8:30
Rasmus
14.3k14479
14.3k14479
asked May 16 '14 at 19:26
user151465
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1 Answer
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$C^*$-Algebras play a mayor role in many versions of spectral theorems for unbounded hermitian operators, for example the famous Spectral Theorems by Neumann or Gelfand. In an abstract sense, spectral theorems basically say, that you can find a representation of a commutative $C^*$-Algebra on Hilbert Spaces as multiplication operators.
This is of course of mayor importance for PDE theory, as you can reduce many PDEs to Eigenvalue Problems of hermitian Operators. For example, if you try separation with $Psi(vec{x},t)=exp{(-frac{i}{hbar}Ecdot t)}cdotpsi(vec{x})$ on the Schrödinger Equation, you get the Eigenvalue Problem $Hpsi=Epsi$, which is known as the stationary Schrödinger Equation, which you can obviously solve using spectral methods.
For more general reasons, $C^*$-Algebras also play a mayor role in Quantum Mechanics and Quantum Field Theory (expecially Axiomatic QFT) as Quantum Mechanics is mathematically nothing else than a very big Eigenvalue Problem. This may be also an interesting thing to know.
answered May 16 '14 at 21:45
DanielDaniel
601413
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$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
1
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
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– Michael
May 22 '14 at 20:57
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1 Answer
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$begingroup$
$C^*$-Algebras play a mayor role in many versions of spectral theorems for unbounded hermitian operators, for example the famous Spectral Theorems by Neumann or Gelfand. In an abstract sense, spectral theorems basically say, that you can find a representation of a commutative $C^*$-Algebra on Hilbert Spaces as multiplication operators.
This is of course of mayor importance for PDE theory, as you can reduce many PDEs to Eigenvalue Problems of hermitian Operators. For example, if you try separation with $Psi(vec{x},t)=exp{(-frac{i}{hbar}Ecdot t)}cdotpsi(vec{x})$ on the Schrödinger Equation, you get the Eigenvalue Problem $Hpsi=Epsi$, which is known as the stationary Schrödinger Equation, which you can obviously solve using spectral methods.
For more general reasons, $C^*$-Algebras also play a mayor role in Quantum Mechanics and Quantum Field Theory (expecially Axiomatic QFT) as Quantum Mechanics is mathematically nothing else than a very big Eigenvalue Problem. This may be also an interesting thing to know.
answered May 16 '14 at 21:45
DanielDaniel
601413
$endgroup$
$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
1
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
$endgroup$
– Michael
May 22 '14 at 20:57
add a comment |
$begingroup$
$C^*$-Algebras play a mayor role in many versions of spectral theorems for unbounded hermitian operators, for example the famous Spectral Theorems by Neumann or Gelfand. In an abstract sense, spectral theorems basically say, that you can find a representation of a commutative $C^*$-Algebra on Hilbert Spaces as multiplication operators.
This is of course of mayor importance for PDE theory, as you can reduce many PDEs to Eigenvalue Problems of hermitian Operators. For example, if you try separation with $Psi(vec{x},t)=exp{(-frac{i}{hbar}Ecdot t)}cdotpsi(vec{x})$ on the Schrödinger Equation, you get the Eigenvalue Problem $Hpsi=Epsi$, which is known as the stationary Schrödinger Equation, which you can obviously solve using spectral methods.
For more general reasons, $C^*$-Algebras also play a mayor role in Quantum Mechanics and Quantum Field Theory (expecially Axiomatic QFT) as Quantum Mechanics is mathematically nothing else than a very big Eigenvalue Problem. This may be also an interesting thing to know.
answered May 16 '14 at 21:45
DanielDaniel
601413
$endgroup$
$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
1
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
$endgroup$
– Michael
May 22 '14 at 20:57
add a comment |
$begingroup$
$C^*$-Algebras play a mayor role in many versions of spectral theorems for unbounded hermitian operators, for example the famous Spectral Theorems by Neumann or Gelfand. In an abstract sense, spectral theorems basically say, that you can find a representation of a commutative $C^*$-Algebra on Hilbert Spaces as multiplication operators.
This is of course of mayor importance for PDE theory, as you can reduce many PDEs to Eigenvalue Problems of hermitian Operators. For example, if you try separation with $Psi(vec{x},t)=exp{(-frac{i}{hbar}Ecdot t)}cdotpsi(vec{x})$ on the Schrödinger Equation, you get the Eigenvalue Problem $Hpsi=Epsi$, which is known as the stationary Schrödinger Equation, which you can obviously solve using spectral methods.
For more general reasons, $C^*$-Algebras also play a mayor role in Quantum Mechanics and Quantum Field Theory (expecially Axiomatic QFT) as Quantum Mechanics is mathematically nothing else than a very big Eigenvalue Problem. This may be also an interesting thing to know.
answered May 16 '14 at 21:45
DanielDaniel
601413
$endgroup$
$C^*$-Algebras play a mayor role in many versions of spectral theorems for unbounded hermitian operators, for example the famous Spectral Theorems by Neumann or Gelfand. In an abstract sense, spectral theorems basically say, that you can find a representation of a commutative $C^*$-Algebra on Hilbert Spaces as multiplication operators.
This is of course of mayor importance for PDE theory, as you can reduce many PDEs to Eigenvalue Problems of hermitian Operators. For example, if you try separation with $Psi(vec{x},t)=exp{(-frac{i}{hbar}Ecdot t)}cdotpsi(vec{x})$ on the Schrödinger Equation, you get the Eigenvalue Problem $Hpsi=Epsi$, which is known as the stationary Schrödinger Equation, which you can obviously solve using spectral methods.
For more general reasons, $C^*$-Algebras also play a mayor role in Quantum Mechanics and Quantum Field Theory (expecially Axiomatic QFT) as Quantum Mechanics is mathematically nothing else than a very big Eigenvalue Problem. This may be also an interesting thing to know.
answered May 16 '14 at 21:45
DanielDaniel
601413
answered May 16 '14 at 21:45
DanielDaniel
601413
answered May 16 '14 at 21:45
DanielDaniel
601413
answered May 16 '14 at 21:45
DanielDaniel
601413
601413
$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
1
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
$endgroup$
– Michael
May 22 '14 at 20:57
add a comment |
$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
1
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
$endgroup$
– Michael
May 22 '14 at 20:57
$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
$begingroup$
Could you elaborate on the spectral theorem for unbounded operators? How is that connected to C* algebras? I always thought the elements of a C* algebras correspond to bounded operators on hilbert spaces.
$endgroup$
– Johannes Hahn
May 17 '14 at 16:25
1
1
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
$endgroup$
– Michael
May 22 '14 at 20:57
$begingroup$
That is kind of over-selling $C^*$-algebras. I mean the spectral theorem, really? Sure the $C^*$-formulation is nice, but by no means necessary. Index theory a la Atiyah-Singer is a deeper connection with PDE's, although I am not sure how much PDE people care about that.
$endgroup$
– Michael
May 22 '14 at 20:57
add a comment |
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