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...to a noncommutative setting by studying those ideas on general von Neumann algebras. ...ebras with a faithful normal tracial state, for example finite von Neumann algebras, the notion of conditional expectation is especially useful. ...

{{short description|Metric on the C*-algebras on a fixed Hilbert space}} ...[[Hausdorff distance]] between the [[unit ball]]s of the two C^*-algebras, under the norm-induced metric on the space of all [[bounded operator]]s on ...

...are determined to a large extent by their projections. The idea behind AW*-algebras is to forgot the former, topological, condition, and use only the latter, a Hence an AW*-algebra is a C*-algebras that is at the same time a [[Baer *-ring]]. ...

...ideal <math>\mathfrak{i}</math> of a ring ''R'' can also be called a (von Neumann) '''regular ideal''' if for each element ''x'' of <math>\mathfrak{i}</math> ...Wesley. Reading, Massachusetts .</ref> This article will use "quotient von Neumann regular" to refer to this type of regular ideal. ...

...ert space]]. They were introduced by [[Paul Dirac]] in 1930 and [[John von Neumann]] in 1932. The Dirac–von Neumann axioms can be formulated in terms of a [[C*-algebra]] as follows. ...

===Central simple algebras=== ...neralized version for [[Artinian ring]]s (which include finite-dimensional algebras) appears in {{harv|Isaacs|2009|loc=p.187}}. Given a [[simple module|simple] ...

...Fuglede−Kadison determinant''' of an invertible operator in a finite [[Von Neumann algebra#Factors|factor]] is a positive real number associated with it. It d ...ators in finite factors, it carries over to the case of operators in [[von Neumann algebra]]s with a tracial state (<math>\tau</math>) in the case of which it ...

...[factor representation]] of a [[C*-algebra]], or to the factor for a [[von Neumann algebra]]: the representation <math>\pi </math> of G is isotypical iff <mat ...etween factor representations and minimal [[central projection]] (in a von Neumann algebra).<ref>{{harvnb|Dixmier|1982|loc= Prop. 5.2.7 p.117}}</ref> Two mini ...

...al <math>[0,1]</math>. Von Neumann was motivated by his discovery of [[von Neumann algebra]]s with a dimension function taking a continuous range of dimension ...metry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms. ...

...to a weak-operator continuous linear functional {{overline|ρ}} on the von Neumann algebra Φ(''A'')^&minus;. If ρ is hermitian, or positive, the sam ...eft ideals in Φ(''A'') from the structure theory of ideals for von Neumann algebras, which is relatively much more simple. If ''K'' is a norm-closed left ideal ...

...[[John von Neumann]]'s study of symmetric norms on [[linear algebra|matrix algebras]].<ref name="vN1">{{cite journal | author= J. von Neumann ...

{{short description|Mathematical problem in von Neumann algebra theory}} ...ng problem is related to the existence of microstates. Some results of von Neumann algebra theory can be obtained assuming positive solution to the problem. T ...

| known_for = [[Von Neumann algebra]]s, [[subfactor]]s, [[ergodic theory]] ...March 1953) is a [[Romanian American]] mathematician working on [[operator algebras]]. He is a professor at the [[University of California, Los Angeles]].<ref ...

...ence G|last2=Pedersen|authorlink1=Lawrence G. Brown|first2=Gert K|title=C*-algebras of real rank zero|journal=Journal of Functional Analysis|date=July 1991|vol ...unori|last2=Osaka|first2=Hiroyuki|title=Real Rank of Tensor Products of C*-algebras|journal=Proceedings of the American Mathematical Society|date=July 1995|vol ...

...is the case where <math>\mathcal{A}</math> is a [[Banach algebra#Banach *-algebras|complete normed *-algebra]]. This algebra satisfies the C*-identity (<math> ...irst=Bruce |title=Operator Algebras. Theory of C*-Algebras and von Neumann Algebras |publisher=Springer |location=Berlin/Heidelberg |year=2006 |isbn=3-540-2848 ...

{{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]]. ...

...].) Atiyah defined also {{nowrap|<math>l^2</math>-Betti}} numbers as [[von Neumann dimension]]s of the resulting {{nowrap|<math>l^2</math>-cohomology}} groups | chapter = Elliptic operators, discrete groups and von Neumann algebras ...

...ixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in [[quantum mechanics]] and in [[complex geometry]], whe ===JB* algebras=== ...

...tteli]]. Later, [[George A. Elliott]] gave a complete classification of AF algebras using the ''K''₀ functor whose range consists of [[ordered group ...lgebras, both existence and uniqueness follow from the fact the Murray-von Neumann semigroup of projections in an AF algebra is cancellative. ...

...ely-additive measure]] on its [[Stone's representation theorem for Boolean algebras|Stone dual]] gives a valuation.<ref name=LatticeThry2 />{{rp|252–254}} ...doi:10.1109/tsmcb.2003.818558]]</ref> Metric lattices are also key to von Neumann's construction of the [[continuous geometry|continuous projective geometry] ...