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The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann algebra …

By definition, a von Neumann algebra is a C*‑algebra A that admits a predual, i.e., a Banach space Z such that Z* is isomorphic to the underlying Banach space of A. (We require that isomorphisms in th …

Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-N …

Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples. Using his definition it is unclear how to separate the smooth structure from the metric. …

The traditional mathematical approach to quantum mechanics, as developed by von Neumann, is based on Hilbert spaces and unbounded self-adjoint operators. Another approach, which more closely resembles …

Given any von Neumann algebra $M$, we can define its noncommutative $\def\L{{\cal L}} \L^p$-spaces $\L^p(M)$ for any $\def\C{{\bf C}} p∈\C$ such that $\Re p≥0$. Here I use the notation $\L^p:={\rm L}^ …

The standard reference for such matters is Guichardet's paper Sur la catégorie des algèbres de von Neumann. Bulletin des Sciences mathématiques 90 (1966), 41–64. PDF file: http://math.berkeley.edu/~pa …

First of all, one should mention that not every triple (X,B,μ) (i.e., what is often called a measure space) satisfies the property that its C*-algebra of bounded functions is a von Neumann algebra (= …

Blackadar in his textbook on operator algebras gives a complete classification of norm-closed ideals in factors. See Proposition III.1.7.11.

The tensor product of commutative algebras is exactly their coproduct in the category of commutative algebras. In other words, if A and B are two commutative algebras, then the covariant functor that …

Suppose M is a von Neumann algebra. Denote by L its maximal noncommutative localization, i.e., the Ore localization with respect to the set of all left and right regular elements, i.e., elements whose …

Is there a way to equip every C*-algebra A with a functorial topology such that the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra? Here A** denotes the dual of A* in …

Answers: (i) Yes, if we replace states by weights (not every von Neumann algebra admits a faithful state); (ii) Yes (for all von Neumann algebras); (iii) All of them. Suppose M is an arbitrary von Ne …

First let me note that one does not need additional conditions of diffuseness and separability in the statement of Gelfand-Neumark theorem for commutative von Neumann algebras. In fact, the category o …

The spaces $[0,1]$, $[0,1]^2$, and $S^1$ are all isomorphic as measurable spaces, including their sets of measure 0, as required by the Gelfand-type duality for measurable spaces. For instance, the is …