Trace and Commutators of Measurable Operators Affiliated to a Von Neumann Algebra
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TRACE AND COMMUTATORS OF MEASURABLE OPERATORS AFFILIATED TO A VON NEUMANN ALGEBRA A. M. Bikchentaev
UDC 517.983, 517.986
Abstract. In this paper, we present new properties of the space L1 (M, τ ) of integrable (with respect to the trace τ ) operators affiliated to a semifinite von Neumann algebra M. For self-adjoint τ -measurable operators A and B, we find sufficient conditions of the τ -integrability of the operator λI − AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ R. Under these conditions, [A, B] = AB − BA ∈ L1 (M, τ ) and τ ([A, B]) = 0. For τ -measurable operators A and B = B 2 , we find conditions that are sufficient for the validity of the relation τ ([A, B]) = 0. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L1 (M, τ ) if and only if I − A, I − U ∈ L1 (M, τ ). For a τ -measurable operator A, we present estimates of the trace of the autocommutator [A∗ , A]. Let self-adjoint τ -measurable operators X ≥ 0 and Y be such that [X 1/2 , Y X 1/2 ] ∈ L1 (M, τ ). Then τ ([X 1/2 , Y X 1/2 ]) = it, where t ∈ R and t = 0 for XY ∈ L1 (M, τ ). Keywords and phrases: Hilbert space, linear operator, von Neumann algebra, normal semifinite trace, measurable operator, integrable operator, commutator, autocommutator. AMS Subject Classification: 47C15, 46L51
CONTENTS 1. 2. 3. 4.
Introduction . . . . . . . Notation and Definitions Lemmas and Examples . Basic Results . . . . . . References . . . . . . . .
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Introduction
Let a von Neumann algebra M of operators act in a Hilbert space H and τ be an exact, normal, semifinite trace on M. We state new properties of the space L1 (M, τ ) of integrable operators affiliated to the algebra M. For an operator X ∈ L1 (M, τ ), we examine conditions under which τ (X) ∈ R or τ (X) = 0. For self-adjoint τ -measurable operators A and B, we find sufficient conditions of the integrability of the operator λI − AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ R. Under these conditions, the commutator [A, B] = AB − BA belongs to L1 (M, τ ) and τ ([A, B]) = 0 (see Theorems 4.1 and 4.2 and Propositions 4.1–4.4). For τ -measurable operators A and B = B 2 , we find conditions sufficient for the validity of the relation τ ([A, B]) = 0 (Theorem 4.3). Item (ii) of Theorem 4.3 is a generalization of [6, Theorem 2.26]. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L1 (M, τ ) if and only if I − A, I − U ∈ L1 (M, τ ) (Theorem 4.5). For a τ -measurable operator A, we find estimates of the trace of autocommutator [A∗ , A] (Corollary 4.4 and Theorem 4.7). Let self-adjoint, τ -measurable operators X ≥ 0
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