application of c*algebras to PDEs

Question

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.

Answer 1

$C^*$-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 $\Psi(\vec{x},t)=\exp{(-\frac{i}{\hbar}E\cdot t)}\cdot\psi(\vec{x})$ on the Schrödinger Equation, you get the Eigenvalue Problem $H\psi=E\psi$, 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.