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Every commutative von Neumann algebra is isomorphic to L ∞ (X) for some measure space (X, μ) and conversely, for every σ-finite measure space X, the *-algebra L ∞ (X) is a von Neumann algebra. Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C*-algebras is sometimes ...
In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949. [163] [164] Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly ...
A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the discrete topology. Note that for discrete groups, C c (G) is the same thing as the complex group ring C[G]. The importance of the group algebra is that it captures the unitary representation theory of G as shown in the ...
The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces.Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L ∞ (X) for some standard measure space (X, μ) and conversely, for every standard measure space X, L ∞ (X) is a von Neumann algebra.
In set theory, several ways have been proposed to construct the natural numbers. These include the representation via von Neumann ordinals, commonly employed in axiomatic set theory, and a system based on equinumerosity that was proposed by Gottlob Frege and by Bertrand Russell.
This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which is a type II 1 factor.
The modular operator is trivial and the corresponding von Neumann algebra is a direct sum of type I and type II von Neumann algebras. Examples: If M is a von Neumann algebra acting on a Hilbert space H with a cyclic separating unit vector v , then put A {\displaystyle {\mathfrak {A}}} = Mv and define ( xv )( yv ) = xyv and ( xv ) ♯ = x * v .
Then Z(A) is an Abelian von Neumann algebra. Example. The center of L(H) is 1-dimensional. In general, if A is a von Neumann algebra, if the center is 1-dimensional we say A is a factor. When A is a von Neumann algebra whose center contains a sequence of minimal pairwise orthogonal non-zero projections {E i} i ∈ N such that