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...es of the same size as the original. This was proved in 1929 by [[John von Neumann]], assuming the [[axiom of choice]]. It is based on the earlier [[Banach–Ta .... This would make creating two unit squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions was th ...

In mathematics, specifically in [[functional analysis]], a [[Family of sets|family]] <math>\mathcal{G}</math> of subsets a [[topological vector space]] (TVS) ...is any collection of subsets of <math>X</math> then the smallest saturated family containing <math>\mathcal{S}</math> is called the {{em|saturated hull}} of ...

{{Short description|Identifies the commutant of a specific von Neumann algebra}} ...Neumann algebra]] acting on a [[Hilbert space]] in the presence of a [[Von Neumann algebra#Weights, states, and traces|trace]]. ...

...other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism ...generates is associative and hence the self-adjoint part of an Abelian von Neumann algebra. In particular ''a'' can be approximated in norm by linear combinat ...

...algebra <math>\mathfrak{M}</math> of operators on a Hilbert space is a von Neumann algebra if and only if <math>\mathfrak{M}'' = \mathfrak{M}</math>.}} {{defn|1=A [[factor (functional analysis)|factor]] is a von Neumann algebra with trivial center.}} ...

...ctor bornology on <math>X</math> called the {{em|'''[[John von Neumann|von Neumann]] bornology of <math>X</math>'''}}, the {{em|'''usual bornology'''}}, or si ...cal vector space topology on <math>LB(T, \mathbb{K})</math> defined by the family of seminorms <math>\left\{ p_{B} : B \in \mathcal{B} \right\}</math> is cal ...

...lgebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative. The counterpart of simple AF C*-algebras in the [[von Neumann algebra]] world are the hyperfinite factors, which were classified by [[Ala ...

...frac{1}{N}Tr</math>. Even more generally, <math>A</math> could be a [[von Neumann algebra]] and <math>\phi</math> a state on <math>A</math>. A final example Let <math>\{A_i : i\in I\}</math> be a family of unital subalgebras of <math>A</math>. ...

...', in [[information theory]], is the smallest of the [[Rényi entropy|Rényi family]] of entropies, corresponding to the [[Rényi entropy#Min-entropy|most conse ...th the classical Shannon entropy and its quantum generalization, the [[von Neumann entropy]], one can define a conditional version of min-entropy. The condit ...

..., <math>H(A|B)\,=\,H(AB)-H(B) \, .</math> with <math>H(A)</math> the [[Von Neumann entropy]], <math>H(A):=-Tr\rho_A\log\rho_A</math>. It thus provides an ope |title=The mother of all protocols: restructuring quantum information's family tree|journal=Proc. R. Soc. A | volume=465 |pages=2537–2563|year=2009|issue= ...

...ics]] and [[mathematics]] such as [[J. Robert Oppenheimer]] and [[John von Neumann]], and was acquainted with [[Albert Einstein]]. Nussbaum was born to a [[Jewish]] family in [[Rheydt]], a borough of the [[Germany|German]] city [[Mönchengladbach]] ...

...''' For functions ''k''(''n''), ''ε''(''n''), ''d''(''n''), ''m''(''n'') a family Ext&nbsp;=&nbsp;{Ext_''n''} of functions ===Von Neumann extractor=== ...

...ions Research and the Management Sciences|INFORMS]] 1998 <br /> [[John von Neumann Theory Prize]] of [[Institute for Operations Research and Management Scienc ...m the Rhineland-Pfaltz region of Germany in 1728. Soon the spelling of the family name evolved, resulting in Rockafellar, Rockefeller, and many other version ...

...phi;''( ''x''_''&alpha;'') for every bounded increasing directed family of positive operators. Any normal trace on <math>L^{1,\infty}(H)</math> is [[Category:Von Neumann algebras]] ...

...analogy for the three terms are used – ''S''(''ρ_A'') the [[von Neumann entropy]], ''S''(''ρ'') the [[joint quantum entropy]] and ''S''(''ρ<sub>A</ ...on to the [[von Neumann entropy#Properties|strong subadditivity of the von Neumann entropy]].<ref name="madhok-datta-2011">Vaibhav Madhok, Animesh Datta: ''Ro ...

...at the [[University of California at Los Angeles]] (UCLA), and in 1979 the family relocated to Los Angeles. Rita Brickman Effros received her Ph.D. in immuno ...ebra]]s on a separable [[Hilbert space]], the Direct Disintegration of von Neumann algebras and/or representations of <math>C^*</math>-algebras, and the Opera ...

| title = Uber die Verbreiterung von Spektrallinien |author2=J. von Neumann ...

...|''n'' ≥ 3}} it contains a [[free group]] with two generators. [[John von Neumann]] studied the properties of the group of equivalences that make a paradoxic ...e needed in the Banach–Tarski paradox, this led to the long-standing [[von Neumann conjecture]], which was disproved in 1980. ...

...g a [[Thermal reservoir|heat bath]]). Algorithmic cooling is the name of a family of algorithms that are given a set of qubits and purify (cool) a subset of ...bypass the Shannon bound). Such an environment can be a heat bath, and the family of algorithms which use it is named "heat-bath algorithmic cooling".<ref na ...

.... The most famous example of a utility representation theorem is the [[Von Neumann–Morgenstern utility theorem]], which shows that any [[rational agent]] has ...has a continuous utility function, if and only if there exists a countable family E of separable systems on ''X'' such that, for all pairs <math>A\succ B</ma ...