Disintegration theory for von Neumann algebras

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Neumann algebras intensively in the 1930s and 1940s. The background for disintegration theory is the paper On rings of operators. Reduction theory written by von Neumann in 1949. It is not hard to produce examples of von Neumann algebras which are not direct sums of factors. The only abelian factor is the algebra of complex numbers C (up to isomorphism), and thus any direct sum of abelian factors is of the form ⊕ i∈I C. Such abelian algebras will always minimal projections (corresponding to each copy of C). It is however easy to come up with abelian algebras with no minimal projections. An example is the abelian von Neumann algebra L ∞ ([0, 1], λ) viewed as multiplication operators on the Hilbert space L 2 ([0, 1], λ), where λ denotes the usual Lebesgue measure. The line of thought behind this example is that the measure space ([0, 1], λ) is “continuous” in some sense, where as the index set I is discrete. The continuity property is due to the fact that the measure space ([0, 1], λ) has no atoms, i.e. any set of positive measure has a subset of strictly smaller, positive measure. Along the lines of type decomposition, we will need the following theorems, which will not be proved here. Proofs can be found in [K&R II]. Theorem 0.2. If M is a von Neumann algebra, then the type of the commutant (I, II or III) is the same as the type of M. Theorem 0.3 (Type decomposition). If M is a von Neumann algebra acting on a Hilbert space H, there are pairwise orthogonal central projections P n (where n ≤ dim H), P II1 , P II∞ and P III with sum I, maximal with respect to the properties that MP n is of type I n or P n = 0, MP II1 is of type II 1 or P II1 = 0, MP II∞ is of type II ∞ or P II∞ = 0 and MP III is of type III of P III = 0. Every type I von Neumann algebra can be decomposed as a direct sum of algebras of type I m . It can also be decomposed as a direct sum of algebras of type I m with commutants of type I n . Notation. Throughout the project the set of natural numbers (excluding zero), rational, real and complex numbers are denoted by N, Q, R and C, respectively. The notation B(H) will mean the bounded operators on a Hilbert space H. iv
CHAPTER 1 The commutant of an abelian von Neumann algebra 1.1 Maximal abelian algebras acting on separable spaces The following section describes all maximal abelian von Neumann algebras acting on separable Hilbert spaces up to unitary equivalence. First we consider some examples, and then we move on to show that those are the only ones. Example 1.1. Let n be a natural number or ℵ 0 , and let S n denote a set with n elements. The bounded functions l ∞ (S n ) act on the separable Hilbert space l 2 (S n ) as multiplication operators M f . They form a maximal abelian von Neumann algebra A n . The projections correspond to the characteristic functions of subsets of S n , and the ordering on the projections correspond to the natural ordering of the subsets of S n by inclusion. Thus the minimal projections correspond to the single point sets, and A n contains exactly n of those. Each M f ∈ A n has the form ∑ y∈S n f (y)M χ{y} where χ {y} is the characteristic function of {y}, and the sum is convergent in the strong operator topology. It follows that the minimal projections generate A n as a von Neumann algebra. Notice the pleasant rule A n ⊕ A m ≃ A n+m for any n, m = 1, 2, . . . , ℵ 0 . Example 1.2. Consider the separable Hilbert space L 2 ([0, 1]) with the Lebesgue measure. The essentially bounded, measurable functions L ∞ ([0, 1]) act on L 2 ([0, 1]) as multiplication operators, and it is well-known that the algebra of multiplication operators A c is a maximal abelian von Neumann subalgebra of the bounded operators on L 2 ([0, 1]) (see for instance chap. 4 in [Douglas]). The algebra A c has no minimal projections, because the Lebesgue measure has no atoms (i.e. every subset of positive measure has a subset of smaller, positive measure). As remarked in the introduction this should be thought of as a kind of continuity property as opposed to the preceding example. This explains the suffix c. Of course, direct sums of the examples above yield more examples. We have now produced examples A n , A c , A n ⊕ A c (1 ≤ n ≤ ℵ 0 ) of maximal abelian von Neumann algebras acting on separable Hilbert spaces. None of these are ∗ -isomorphic, since A n has precisely n minimal projections and is generated by them, A c has no minimal projections, and A n ⊕A c has precisely n minimal projections, but is not generated by them. The following theorem tells us that these are the only examples. 1
Page 1: D E P A R T M E N T O F M A T H E M
Page 4 and 5: Abstract The main theorem is the ce
Page 8 and 9: 1.1 Maximal abelian algebras acting
Page 10 and 11: 1.1 Maximal abelian algebras acting
Page 12 and 13: 1.2 The commutant of an abelian von
Page 14 and 15: 2.1 Disintegration of Hilbert space
Page 16 and 17: 2.1 Disintegration of Hilbert space
Page 18 and 19: 2.2 Decomposable operators Now we r
Page 20 and 21: 2.2 Decomposable operators Defining
Page 22 and 23: 2.2 Decomposable operators From Lem
Page 24 and 25: 2.3 Decomposable representations Ex
Page 26 and 27: 2.4 Decomposable von Neumann algebr
Page 28 and 29: 2.5 Decompositions and the commutan
Page 36 and 37: 3.1 Decomposition relative to an ab
Page 38 and 39: 3.2 Type of the components Theorem
Page 40 and 41: A.2 Measure theoretic technicalitie
Page 42 and 43: A.3 Polish spaces is countable and
Page 44 and 45: A.4 Analytic sets Remark A.15. The
Page 46 and 47: A.4 Analytic sets For n ∈ N consi
Page 48 and 49: A.5 Measurable selections form a ba

Disintegration theory for von Neumann algebras

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