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Quantum Extensions of Ordinary Maps

Mathematical Physics & Probability

Speaker: Andre Kornell, UC Davis
Location: 3106 MSB
Start time: Thu, Nov 29 2018, 4:10PM

Noncommutative mathematics views unital C*-algebras to be a quantum generalization of compact Hausdorff spaces. In this context, unital C*-algebras may be termed quantum compact Hausdorff spaces. Recent research in quantum information theory has stimulated interest in families of maps indexed by such a quantum compact Hausdorff space.

Some extension problems that are unsolvable in the classical setting become solvable in the quantum setting. In other words, some maps admit no family of extensions indexed by an ordinary nonempty compact Hausdorff space, but do admit a family of extensions indexed by a quantum nonempty compact Hausdorff space. For example, there exist loops in the real projective plane and the figure of eight that are not nullhomotopic in the ordinary sense, but which do admit nonempty quantum families of extensions to the unit disk.

This notion of quantum nullhomotopy degenerates in the presences of infinite continuous quantum families. By an application of Kuiper's theorem, every loop admits an infinite continuous quantum family of extensions to the unit disk.