C*-Independence andW*-Independence of von Neumann Algebras

Hamhalter, Jan

doi:10.1002/1522-2616(200206)239:1<146::aid-mana146>3.0.co;2-o

Cited by 12 publications

(11 citation statements)

References 17 publications

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“…If one can extend any pair of vector states to a normal state, then, using the fact that every normal state is a σ-convex combination of vector states, we see that there is a normal state on B( 2 (Γ)) such that its restriction to W * λ (Γ) gives a prescribed vector state and its restriction to W * (Γ) gives a prescribed normal state. Using these arguments once more we see that W * -independence is equivalent to the existence of a common normal extension for all pairs of vector states (see also [15]). Let us now fix unit vectors x, y ∈ 2 (Γ).…”

Section: Independence Of Reduced Group Algebrasmentioning

confidence: 88%

“…This condition is forced by the existence of a faithful product state over the pair A and B provided that A and B commute [12], or by the existence of a separating family of states with the product property without assuming mutual commutativity of A and B (see [5] for more detailed analysis). In the category of von Neumann algebras the counterpart of the C * -independence is called the W * -independence and it is defined as follows: von Neumann subalgebras M and N in a von Neumann algebra F are called W * -independent if for each pair of normal states ϕ and ψ on M and N , respectively, there is a normal state on F extending both ϕ and ψ. W * -independence always implies C * -independence, the reverse implication is not true in general (see [15] for more details). However, a remarkable theorem of Florig and Summers [12] says that if F above is σ-finite and M and N are C * -independent and mutually commute, then M and N are W * -independent.…”

Section: Introductionmentioning

confidence: 98%

“…However, if in addition M is hyperfinite, then the pair M, M is C * -independent in the product sense as M is semidiscrete (see [6]). (For further characterizations of independence conditions and their interrelations we refer the reader to [14][15][16]26]. )…”

Section: Introductionmentioning

confidence: 99%

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Independence of group algebras

Conti

Hamhalter

2010

Mathematische Nachrichten

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It is shown that major independence conditions for left and right group operator algebras coincide. If Γ is a discrete ICC group, then the reduced left and right group algebras W * λ (Γ) and W * (Γ) are W * -independent. These algebras are moreover independent in the product sense if, and only if, Γ is amenable. If A and B are subgroups of Γ, then the left and right reduced group (sub)algebras W * λ (A) and W * (B) are W * -independent provided that any of the following two conditions is satisfied: (i) A and B have trivial intersection; (ii) A or B is ICC. The results indicate an interplay between intrinsic group-theoretic properties and independence of the corresponding group algebras that can be further exploited. New examples of W * -independent von Neumann algebras arising from groups are generated.

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Section: Independence Of Reduced Group Algebrasmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 98%

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Independence of group algebras

Conti

Hamhalter

2010

Mathematische Nachrichten

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“…W * -independence implies C * -independence [71,21]. If M ⊂ N ′ , then these notions are equivalent [21], but W * -independence is strictly stronger when the algebras do not mutually commute [28]. If M ⊂ N ′ and (M, N ) is C * -independent, then the (not necessarily normal) joint extension can be chosen to be a product state [60]:…”

Section: Independencementioning

confidence: 99%

Quantum probability theory

Rédei

Summers

2007

Studies in History and Philosophy of Science Part B: Studies in

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The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

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“…Recent papers that discuss various forms of independence in operator algebras include [3,4] and [6,7]. The comprehensive survey [12] is a main source of information on both mathematical and physical aspects.…”

Section: Introductionmentioning

confidence: 99%

C*-Independence, Product States and Commutation

Bunce

Hamhalter

2004

Ann. Henri Poincaré

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Let D be a unital C * -algebra generated by C * -subalgebras A and B possessing the unit of D. Motivated by the commutation problem of C * -independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Roos's theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation π of D such that π(A) and π(B) commute and π is faithful on both A and B; (iii) A ⊗ min B is canonically isomorphic to a quotient of D.The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C * -algebras must commute and D be their minimal tensor product.

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C-Independence andW-Independence of von Neumann Algebras

Cited by 12 publications

References 17 publications

Independence of group algebras

Independence of group algebras

Quantum probability theory

C*-Independence, Product States and Commutation

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