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The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above. The crossed product of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra.
5 Tensor product of distributions. ... the spaces of test functions and distributions are ... any (von Neumann) bounded subset of + is a relatively ...
The formalism of density operators and matrices was introduced in 1927 by John von Neumann [29] and independently, but less systematically, by Lev Landau [30] and later in 1946 by Felix Bloch. [31] Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.
Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming. [274]
The state space of the entire quantum system is then the tensor product of the spaces for the two parts. :=. Let ρ AB be a density matrix acting on states in H AB. The von Neumann entropy of a density matrix S(ρ), is the quantum mechanical analogy of the Shannon entropy.
Von Neumann's traditional definition simply takes the "obvious" tensor product: to compute , first collect all simple tensors of the form such that ‖ ‖ <. The latter describes a pre-inner product through the polarization identity , so take the closed span of such simple tensors modulo that inner product's isotropy subspaces.
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
5.5 Tensor products of distributions. ... are called test functions on U and ... any (von Neumann) bounded subset of + is a relatively ...