Welcome to Analysis

Analysis is that branch of mathematics which evolved from the notions of approximations, limits and continuity of real- and complex-valued functions. While the fundamental theorem of calculus remains to this day one of the central pillars of mathematics, modern analysis deals with the above concepts in broader and more abstract settings such as those of topological groups and algebras.

Events

Thursday, February 29, 2024 4:30 pm - 5:30 pm EST

Derivations for symmetric quantum Markov semigroups

Matthijs Vernooij, TU Delft

Quantum Markov semigroups describe the time evolution of the operators in a von Neumann algebra corresponding to an open quantum system. Of particular interest are so-called symmetric semigroups. Given a faithful state, one can define the GNS- and KMS-inner product on the von Neumann algebra, and a semigroup is GNS- or KMS-symmetric if it is self-adjoint w.r.t. the inner product. GNS-symmetry implies KMS-symmetry, and both coincide if the state is a trace. It was shown in 2003 that the generator of a tracially symmetric quantum Markov semigroup can be written as the 'square' of a derivation, i.e. d* after d, where d is a derivation to a Hilbert bimodule. This result has proven to be very influential in many different directions. In this talk, we will look at this problem in the case that our state is not tracial. We will start by discussing how a computer can be used to decide whether such a derivation exists in finite dimensions, and work our way up to a general result on KMS-symmetric quantum Markov semigroups. This is joint work with Melchior Wirth.

This seminar will be held both online and in person: