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Magnetic Pseudodifferential Super Operators — At the Intersection of Functional Analysis and Operator Algebras

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Max Lein
Tohoku University
Thu, 21/04/2022 - 15:00 - 16:00
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Hybrid talk.
For zoom details contact Christian Kaiser (kaiser@mpim...)

Super operators like Liouville or Lindblad operators are operators that map operators onto operators. They can naturally be understood as maps between non-commutative L^p spaces and Sobolev spaces, which are generalizations of p-Schatten classes with respect to other traces. The trace plays the role of the integral and one can introduce derivations, which give a notion of regularity.

At this intersection of functional analysis and operator algebras, Giuseppe De Nittis and I developed a framework to make linear response theory rigorous, where a system is driven out of equilibrium and one wants to justify a “Taylor expansion” of the current in the perturbation parameters. This has the advantage of accommodating disorder and works with bounded and unbounded operators on the continuum and discrete alike. The downside is that we had had to make a number of rather technical assumptions, some of which are likely hard to verify for concrete operators. Among other things they are necessary to ensure the existence and well-behavedness of operator products and commutators.

This motivated Gihyun Lee and I to develop a magnetic pseudodifferential calculus for super operators. The idea is that since products and commutators of two pseudodifferential operators are again pseudodifferential, it would free us from having to prove their existence “by hand” and hopefully simplify the assumptions for pseudodifferential (super) operators. I will explain how this calculus is set up and what questions we intend to answer with it in future works.

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