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A von Neumann algebra is an associative Banach algebra over $\mathbb{C}$, which also has an anti-linear anti-involution * such that $||a^*a|| = ||a||^2$, and which also has a predual as a Banach space. In context, you can think of it as a non-associative non-commutative algebra with a certain semisimple-like property and certain fairly strong analytic closure properties.

Now, you can have a non-commutative superalgebra, but this is a somewhat thin combination, because the associativity axiom of an algebra (and in fact every axiom for a von Neumann algebra) does not use the switching map $v \otimes w \mapsto w \otimes v$ or its superized version $v \otimes w \mapsto (-1)^{(\deg v)(\deg w)} w \otimes v$. A supercommutative algebra is not usually a commutative algebra, a Lie seperalgebra superalgebra is not usually a Lie algebra, and a Hopf superalgebra is not usually a Hopf algebra; all of these objects have axioms that use the switching map. But an associative superalgebra is an associative algebra and a von Neumann superalgebra is a von Neumann algebra.

On the other hand, in quantum physics one is often interested in a classical limit which is commutative, or in the supersymmetry context, supercommutative. It is an interesting fact that you can make a commutative von Neumann algebra, which is then a model of classical probability. But you can't make a nontrivially supercommutative von Neumann algebra, because it doesn't have the semisimple-like properties of a von Neumann algebra. However, von Neumann algebra axioms really are necessary for the quantum probability model. So the conventional thing to do is to embed the supercommutative algebra that exists in a theory such as supersymmetry in a von Neumann algebra, even though it is not a von Neumann subalgebra. Or, you could say that supersymmetry (if you accept it) and quantum probability are two ultimately different reasons that classical probability has to be changed. Supersymmetry can be viewed as more of a geometric reason than a probabilistic reason.

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A von Neumann algebra is an associative Banach algebra over $\mathbb{C}$, which also has an anti-linear anti-involution * such that $||a^*a|| = ||a||^2$, and which also has a predual as a Banach space. In context, you can think of it as a non-associative algebra with a certain semi-simpleness semisimple-like property and certain fairly strong analytic closure properties.

Now, you can have a non-commutative superalgebra, but this is a somewhat thin combination, because the associativity axiom of an algebra (and in fact every axiom for a von Neumann algebra) does not use the switching map $v \otimes w \mapsto w \otimes v$ or its superized version $v \otimes w \mapsto (-1)^{(\deg v)(\deg w)} w \otimes v$. A supercommutative algebra is not usually a commutative algebra, a Lie seperalgebra is not usually a Lie algebra, and a Hopf superalgebra is not usually a Hopf algebra; all of these objects have axioms that use the switching map. But an associative superalgebra is an associative algebra and a von Neumann superalgebra is a von Neumann algebra.

On the other hand, in quantum physics one is often interested in a classical limit which is commutative, or in the supersymmetry context, supercommutative. It is an interesting fact that you can make a commutative von Neumann algebra, which is then a model of classical probability. But you can't make a nontrivially supercommutative von Neumann algebra, because it doesn't have the semisimple-like properties of a von Neumann algebra. However, von Neumann algebra axioms really are necessary for the quantum probability model. So the conventional thing to do is to embed the supercommutative algebra that exists in a theory such as supersymmetry in a von Neumann algebra, even though it is not a von Neumann subalgebra. Or, you could say that supersymmetry (if you accept it) and quantum probability are two ultimately different reasons that classical probability has to be changed. Supersymmetry can be viewed as more of a geometric reason than a probabilistic reason.

Post Undeleted by Greg Kuperberg
Post Deleted by Greg Kuperberg
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