PlanetPhysics/Quantum Groups and Von Neumann Algebras

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Hilbert spaces, Von Neumann algebras and Quantum Groups

John von Neumann introduced a mathematical foundation for [[../QuantumParadox/|quantum mechanics]] in the form of [[../WeakHopfCAlgebra/|$W^{*}$-algebras]] of (quantum) bounded [[../QuantumOperatorAlgebra4/|operators]] in a (quantum:= presumed separable , i.e. with a countable basis) [[../NormInducedByInnerProduct/|Hilbert space]] $H_S$. Recently, such [[../WeakHopfCAlgebra2/|von Neumann algebras, $W^{*}$]] and/or (more generally) [[../VonNeumannAlgebra2/|C*-algebras]] are, for example, employed to define \htmladdnormallink{locally compact quantum groups $CQ{G}_{lc}$}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html} by equipping such [[../WeakHopfCAlgebra2/|algebras with a co-associative multiplication]] and also with associated, both left-- and right-- [[../HigherDimensionalQuantumAlgebroid/|Haar measures]], defined by two semi-finite normal weights .

Remark on Jordan-Banach-von Neumann (JBW) algebras, JBWA

A Jordan--Banach algebra (a JB--algebra for short) is both a real Jordan algebra and a [[../NormInducedByInnerProduct/|Banach space]], where for all Failed to parse (unknown function "\bR"): {\displaystyle S, T \in \mathfrak A_{\bR}} , we have \bigbreak \bigbreak \bigbreak Failed to parse (syntax error): {\displaystyle \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. } \bigbreak \bigbreak \bigbreak

A JLB--algebra is a $JB$--algebra Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} together with a Poisson bracket for which it becomes a Jordan--Lie algebra $JL$ for some ${\hslash }^2\ge 0$~. Such JLB--algebras often constitute the real part of several widely studied [[../OrthomodularLatticeTheory/|complex associative algebras]]. For the purpose of [[../MoyalDeformation/|quantization]], there are fundamental [[../Bijective/|relations]] between \htmladdnormallink{${𝔄}^{sa}$, JLB and Poisson algebras}{http://planetphysics.us/encyclopedia/JordanBanachAndJordanLieAlgebras.html}. \bigbreak A JB--algebra which is monotone complete and admits a separating set of normal sets is called a JBW-algebra .

These appeared in the [[../Work/|work]] of von Neumann who developed an \htmladdnormallink{orthomodular lattice theory {http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} of projections on $ℒ(H)$} on which to study [[../LQG2/|quantum logic]]. BW-algebras have the following property: whereas ${𝔄}^{sa}$ is a J(L)B--algebra, the self-adjoint part of a von Neumann algebra is a JBW--algebra.