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A von Neumann approach to quantum metrics

QMAP Seminar

Speaker: Greg Kuperberg, UC Davis
Location: 2112 MSB
Start time: Thu, Dec 5 2013, 4:10PM

I will outline joint work with Nik Weaver, in which we give a definition of a quantum metric space based on von Neumann algebras. Given a von Neumann algebra M with a faithful representation H, a quantum metric on M in our sense is a weak-*-closed, self-adjoint algebra filtration on B(H), whose zero-term is the commutant of M. It turns out that this structure depends only on M and not on the specific choice of H. The construction strictly generalizes the definition of a classical metric on a set, and a metric on a measurable space in the sense of Weaver. Our definition is inspired by, and is suitable for, the theory of quantum error correction in quantum computation. As such it has "tropical" properties. Finally,just as a classical metric on a set defines both continuous and Lipschitz functions, our quantum metric structure yields both a C^*-algebra of uniform elements, and an algebra of Lipschitz elements.