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 quantum mechanics in the form of $W^{*}$-algebras of (quantum) bounded operators in a (quantum:= presumed separable , i.e. with a countable basis) Hilbert space $H_{S}$. Recently, such von Neumann algebras, $W^{*}$ and/or (more generally) C*-algebras are, for example, employed to define \htmladdnormallink{locally compact quantum groups $CQG_{lc}$}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html} by equipping such algebras with a co-associative multiplication and also with associated, both left-- and right-- 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 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 $\hslash ^{2}\geq 0$~. Such JLB--algebras often constitute the real part of several widely studied complex associative algebras. For the purpose of quantization, there are fundamental relations between \htmladdnormallink{${\mathfrak {A}}^{sa}$, 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 of von Neumann who developed an \htmladdnormallink{orthomodular lattice theory {http://planetphysics.us/encyclopedia/OrthomodularLatticeTheory.html} of projections on ${\mathcal {L}}(H)$} on which to study quantum logic. BW-algebras have the following property: whereas ${\mathfrak {A}}^{sa}$ is a J(L)B--algebra, the self-adjoint part of a von Neumann algebra is a JBW--algebra.