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 $$S, T \in \mathfrak A_{\bR}$$, we have \bigbreak \bigbreak \bigbreak $$ \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. $$ \bigbreak \bigbreak \bigbreak

A JLB--algebra is a $$JB$$--algebra $$\mathfrak A_{\bR}$$ 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.