Theorems in functional analysis | Articles containing proofs | Operator theory | Von Neumann algebras
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory. The formal statement of the theorem is as follows: Von Neumann bicommutant theorem. Let M be an algebra consisting of bounded operators on a Hilbert space H, containing the identity operator, and closed under taking adjoints. Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M. This algebra is called the von Neumann algebra generated by M. There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies. It is related to the Jacobson density theorem. (Wikipedia).
David Penneys: Bicommutant categories from multifusion categories
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From playlist Physique mathématique des nombres de Hurwitz pour débutants
Maxim Kazarian - 1/3 Mathematical Physics of Hurwitz numbers
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From playlist Physique mathématique des nombres de Hurwitz pour débutants
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Maxim Kazarian - 2/3 Mathematical Physics of Hurwitz numbers
Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge num
From playlist Physique mathématique des nombres de Hurwitz pour débutants
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Workshop 1 "Operator Algebras and Quantum Information Theory" - CEB T3 2017 - D.Farenick
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History of Science and Technology Q&A (October 20, 2021)
Stephen Wolfram hosts a live and unscripted Ask Me Anything about the history of science and technology for all ages. Find the playlist of Q&A's here: https://wolfr.am/youtube-sw-qa Originally livestreamed at: https://twitch.tv/stephen_wolfram/ Outline of Q&A 0:00 Stream starts 1:41 S
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