Local convergence in measure on semifinite von Neumann algebras

Consider a von Neumann algebra M with a faithful normal semifinite trace τ . We prove that each order bounded sequence of τ -compact operators includes a subsequence whose arithmetic averages converge in τ . We also prove a noncommutative analog of Pratt's lemma for L 1 (M, τ). The results are new even for the algebra M = B(H) of bounded linear operators with the canonical trace τ = tr on a Hilbert space H. We apply the main result to L p (M, τ) with 0 < p ≤ 1 and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of τ -compactness of the dominant.

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“…Consequently, Y − X n ∈ M + 0 and Y − X n ↓ 0 (as n → ∞).Thus, Y − X n τ −→ 0 as n → ∞ by Lemma 3.14 of[23]. Similarly, Y n τ −→ Y as n → ∞.…”

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confidence: 71%

Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators

Бикчентаев

Sabirova

2012

Sib Math J

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“…Hence, S˛t l ! 0 [8]. Repeating the final part of the proof of assertion (i) of Theorem 1, we establish that t h l Ä t o .M/:…”

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confidence: 87%

“…If t w l D t.M; /; then the operation of multiplication in S.M; /; t w l is continuous in the collection of variables. In this case, as shown in [8] (Theorem 4.1), we have t l D t w l ; and M is of finite type. The implication (ii) ) (iii) is established by analogy.…”

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confidence: 90%

“…If M is of finite type, then t w l D t l [8]. Hence that n k P n k Ä S for any k D 1; 2; : : : : Therefore, 0 Ä P n k Ä .n k / 1 S t !…”

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confidence: 90%

“…2 .6), and the topology of convergence in measure generated by the trace is a natural topology that equips these -algebras with the structure of a topological -algebra [3]. If the trace is semifinite but not finite, then one can consider the topologies t l of -local convergence and t w l of weaklylocal convergence in measure [8]. However, in these topologies, the operation of multiplication is not continuous in the collection of variables in the case where M does not have a finite type.…”

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confidence: 99%

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(o)-Topology in *-algebras of locally measurable operators

Muratov

Chilin

2009

Ukr Math J

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We consider the topology t.M/ of convergence locally in measure in the -algebra LS.M/ of all locally measurable operators affiliated to the von Neumann algebra M: We prove that t.M/ coincides with the .o/-topology in LS h .M/ D fT 2 LS.M/W T D T g if and only if the algebra M is -finite and is of finite type. We also establish relations between t.M/ and various topologies generated by a faithful normal semifinite trace on M: Let L 0 . ; †; / be the -algebra of all measurable complex functions defined on a space with complete measure . ; †; / (almost-everywhere equal functions are identified). If . / < 1; then the .o/-topology in L 0 h . ; †; / D ff 2 L 0 . ; †; /W f D f g coincides with the topology of convergence in measure (see, e.g., [1], Chap. III, Sec. 9). In the case where the measure is -finite, the convergence in the .o/-topology t o in L 0 h . ; †; / is equivalent to the convergence locally in measure ; i.e., f˛t o ! f; f˛; f 2 L 0 h . ; †; /; if and only if f˛ A ! f A for all A 2 † with .A/ < 1 [1] (Chap. VI, Sec. 3). If the measure is not -finite but possesses the property of direct sum, then the .o/-topology is essentially stronger than the topology of convergence locally in measure [2] (Chap. V, Secs. 4 and 6). The development of the theory of integration for a faithful normal semifinite trace defined on the von Neumann algebra M led to the necessity of investigation of the -algebra S.M; / of -measurable operators (see, e.g., [3]). This algebra is a filled -subalgebra in the -algebra S.M/ of all measurable operators affiliated to M: The -algebras S.M/ were introduced by Segal in [4] for the description of a "noncommutative" version of the -algebra of measurable complex functions. If M is a commutative von Neumann algebra, then M can be identified with the -algebra L 1 . ; †; / of all essentially bounded measurable complex functions defined on a measurable space . ; †; / with complete measure that possesses the property of direct sum. In this case, the -algebra S.M/ is identified with the -algebra L 0 . ; †; / [4]. The -algebras S.M; / and S.M/ are informative examples of EW -algebras E of closed linear operators affiliated to the von Neumann algebra M that act in the same Hilbert space H as M and whose bounded part E b D E \ B.H/ coincides with M (see [5]), where B.H/ is the -algebra of all bounded linear operators in H: The natural desire to construct the maximum EW -algebra E with E b D M led to the introduction of the -algebra LS.M/ of all locally measurable operators affiliated to M [5]. It was shown in [6] that any EW -algebra E with E b D M is a filled -subalgebra in LS.M/: In the case of the existence of a faithful normal trace on M; all three -algebras LS.M/; S.M/; and S.M; / coincide [7] (Sec. 2 .6), and the topology of convergence in measure generated by the trace is a natural topology that equips these -algebras with the structure of a topological -algebra [3]. If the trace is semifinite but not finite, then one can consider the topologies t l of -local convergence and t w l of weaklylocal...

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