Block projection operators in normed solid spaces of measurable operators

We prove a Hermitian analog of the well-known operator triangle inequality for vonNeumann algebras. In the semifinite case we show that a block projection operator is a linear positive contraction on a wide class of solid spaces of Segal measurable operators. We describe some applications of the results. © 2012 Allerton Press, Inc

On hermitian operators X and Y meeting the condition -Y ≤ X ≤ Y

We obtain a description of all pairs of Hermitian operators X and Y, which satisfy the condition -Y ≤ X ≤ Y. We give the examples of such operator pairs. Each of the presented examples leads us to the new weak majorization for the Hermitian operator pair. It is shown that this inequality does not necessarily imply the inequality {pipe}X{pipe} ≤ ZY Z* for any operator Z, {double pipe}Z{double pipe} ≤ 1. We prove that invertibility of Y follows from invertibility of operators X and A* A for Hermitian operators X and Y, Y ≥ 0 and an arbitrary operator A such that -AY A* ≤ X ≤ AY A*. We discuss one analog of triangle inequality found by the author in one earlier paper for pairs of Hermitian operators. © 2013 Pleiades Publishing, Ltd

Two classes of τ-measurable operators affiliated with a von Neumann algebra

© 2017, Allerton Press, Inc.Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from M and k = 1, 2; if an operator T from P1 admits the bounded inverse T−1, then T−1 lies in P1. We establish some new inequalities for rearrangements of operators from P1. If a τ-measurable operator T is hyponormal and Tn is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators on H

On idempotent τ-measurable operators affiliated to a von Neumann algebra

© 2016, Pleiades Publishing, Ltd.Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 2, and A = An ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μt(A) belong to the set {0} ∪ [‖An−2‖−1, ‖A‖] for all t > 0; 2) either μt(A) ≥ 1 for all t > 0 or there is a t0 > 0 such that μt(A) = 0 for all t > t0. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z2 = 0, ZP = 0, and PZ = Z. Here, if Q ∈ Lp(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L1(M, τ) and if A = A3 and A − A2 ∈ M, then τ(A) ∈ R

Concerning the theory of τ-measurable operators affiliated to a semifinite von Neumann algebra

© 2015, Pleiades Publishing, Ltd. Let M be a von Neumann algebra of operators in a Hilbert space H, let τ be an exact normal semifinite trace on M, and let L1(M, τ) be the Banach space of τ-integrable operators. The following results are obtained. If X = X*, Y = Y* are τ-measurable operators and XY ∈ L1(M, τ), then YX ∈ L1(M, τ) and τ(XY) = τ(YX) ∈ R. In particular, if X, Y ∈ B(H)sa and XY ∈ G1, then YX ∈ G1 and tr(XY) = tr(YX) ∈ R. If X ∈ L1(M, τ), then (Formula Presented.). Let A be a τ-measurable operator. If the operator A is τ-compact and V ∈ M is a contraction, then it follows from V* AV = A that V A = AV. We have A = A2 if and only if A = |A*||A|. This representation is also new for bounded idempotents in H. If A = A2 ∈ L1(M, τ), then (Formula Presented.). If A = A2 and A (or A*) is semihyponormal, then A is normal, thus A is a projection. If A = A3 and A is hyponormal or cohyponormal, then A is normal, and thus A = A* ∈ M is the difference of two mutually orthogonal projections (A + A2)/2 and (A2 − A)/2. If A,A2 ∈ L1(M, τ) and A = A3, then τ(A) ∈ R

Ideal F-norms on C*-algebras

© 2015, Allerton Press, Inc. We show that every measure of non-compactness on a W*-algebra is an ideal F-pseudonorm. We establish a criterion of the right Fredholm property of an element with respect to a W*-algebra. We prove that the element −I realizes the maximum distance from a positive element to a subset of all isometries of a unital C*-algebra, here I is the unit of the C*-algebra. We also consider differences of two finite products of elements from the unit ball of a C*-algebra and obtain an estimate of their ideal F-pseudonorms. We conclude the paper with a convergence criterion in complete ideal F-norm for two series of elements from a W*-algebra

Convergence of integrable operators affiliated to a finite von Neumann algebra

© 2016, Pleiades Publishing, Ltd.In the Banach space L1(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L2(M, τ) is introduced, and its main properties are established. A convergence criterion in L2(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L1(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖1≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L1(M, τ) is complemented. The convergence in L2(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained

On additivity of mappings on measurable functions

We prove the additivity of regular l-additive mappings T: [InlineMediaObject not available: see fulltext.] → [0,+∞] of a hereditary cone [InlineMediaObject not available: see fulltext.] in the space of measurable functions on a measure space. Some examples are constructed of non-d-additive l-additive mappings T. The monotonicity of l-additive mappings T: [InlineMediaObject not available: see fulltext.] → [0,+∞] is established. The examples are constructed of nonmonotone d-additive mappings T. Let (S, +) be a commutative cancellation semigroup. Given a mapping T: [InlineMediaObject not available: see fulltext.] → S, we prove the equivalence of additivity and l-additivity. It is shown that a strongly regular 2-homogeneous l-subadditive mapping T is subadditive. All results are new even in case [InlineMediaObject not available: see fulltext.] = L∞ +. © 2014 Pleiades Publishing, Ltd

On operator monotone and operator convex functions

© 2016, Allerton Press, Inc.We establish monotonicity and convexity criteria for a continuous function f: R+ → R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive τ-measurable operator and a positive operator from a von Neumann algebra

The Haagerup problem on subadditive weights on W*-algebras. II

In 1975 U. Haagerup has posed the following question: Whether every normal subadditive weight on a W*-algebra is σ-weakly lower semicontinuous? In 2011 the author has positively answered this question in the particular case of abelian W*-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional W*-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic W*-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic W*-algebras. © 2013 Allerton Press, Inc