The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
A M Bikchentaev - One of the best experts on this subject based on the ideXlab platform.
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Paranormal measurable operators affiliated with a semifinite Von Neumann Algebra. II
Positivity, 2020Co-Authors: A M BikchentaevAbstract:Let $${{\mathcal {M}}}$$ M be a Von Neumann Algebra of operators on a Hilbert space $${\mathcal {H}}$$ H and $$\tau $$ τ be a faithful normal semifinite trace on $$\mathcal {M}$$ M . Let $$t_\tau $$ t τ be the measure topology on the $$*$$ ∗ -Algebra $$S(\mathcal {M},\tau )$$ S ( M , τ ) of all $$\tau $$ τ -measurable operators. We define three $$t_\tau $$ t τ -closed classes $${{\mathcal {P}}}_1$$ P 1 , $${{\mathcal {P}}}_2$$ P 2 and $${{\mathcal {P}}}_3$$ P 3 of $$\tau $$ τ -measurable operators and investigate their properties. The class $${{\mathcal {P}}}_2$$ P 2 contains $${{\mathcal {P}}}_1\cup {{\mathcal {P}}}_3$$ P 1 ∪ P 3 . If a $$\tau $$ τ -measurable operator T is hyponormal, then T lies in $${{\mathcal {P}}}_1\cap {{\mathcal {P}}}_3$$ P 1 ∩ P 3 ; if an operator T lies in $${{\mathcal {P}}}_3$$ P 3 , then $$UTU^*$$ U T U ∗ belongs to $${{\mathcal {P}}}_3$$ P 3 for all isometries U from $${{\mathcal {M}}}$$ M . If a bounded operator T lies in $$\mathcal {P}_1\cup {{\mathcal {P}}}_3$$ P 1 ∪ P 3 then T is normaloid. If an operator $$T\in S(\mathcal {M},\tau )$$ T ∈ S ( M , τ ) is p -hyponormal with $$0
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metrics on projections of the Von Neumann Algebra associated with tracial functionals
Siberian Mathematical Journal, 2019Co-Authors: A M BikchentaevAbstract:Let φ be a positive functional on a Von Neumann Algebra $$\mathscr{A}$$ and let $$\mathscr{A}^{\rm{pr}}$$ be the projection lattice in $$\mathscr{A}$$. Given $$P,Q \in \mathscr{A}^{\rm{pr}}$$, put ρφ(P, Q) = φ(∣P − Q∣) and dφ(P, Q) = φ(P ∨ Q − P ∧ Q). Then ρφ(P, Q) ≤ dφ(P, Q) and ρφ(P, Q) = dφ(P, Q) provided that PQ = QP. The mapping ρφ (or dφ) meets the triangle inequality if and only if φ is a tracial functional. If τ is a faithful tracial functional then ρτ and dτ are metrics on $$\mathscr{A}^{\rm{pr}}$$. Moreover, if τ is normal then ($$\mathscr{A}^{\rm{pr}}$$, ρτ) and ($$\mathscr{A}^{\rm{pr}}$$, dτ) are complete metric spaces. Convergences with respect to ρτ and dτ are equivalent if and only if $$\mathscr{A}$$ is abelian; in this case ρτ = dτ. We give one more criterion for commutativity of $$\mathscr{A}$$ in terms of inequalities.
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Paranormal Measurable Operators Affiliated with a Semifinite Von Neumann Algebra
Lobachevskii Journal of Mathematics, 2018Co-Authors: A M BikchentaevAbstract: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 P _1 and P _2 of τ -measurable operators and investigate their properties. The class P _2 contains P _1. If a τ -measurable operator T is hyponormal, then T lies in P _1; if an operator T lies in P _k, then UTU * belongs to P _k for all isometries U from Mand k = 1, 2; if an operator T from P _1 admits the bounded inverse T ^−1 then T ^−1 lies in P _1. If a bounded operator T lies in P _1 then T is normaloid, T ^ n belongs to P _1 and a rearrangement μt ( T ^ n ) ≥ μt ( T )^ n for all t > 0 and natural n . If a τ -measurable operator T is hyponormal and T ^ n is τ -compact operator for some natural number n then T is both normal and τ -compact. If an operator T lies in P _1 then T 2 belongs to P _1. If M = B ( H ) and τ = tr, then the class P _1 coincides with the set of all paranormal operators on H . If a τ-measurable operator A is q -hyponormal (1 ≥ q > 0) and | A *| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q -cohyponormal) operator is normal. Consider a τ -measurable nilpotent operator Z ≠ 0 and numbers a , b ∈ R . Then an operator Z * Z − ZZ * + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.
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ideal spaces of measurable operators affiliated to a semifinite Von Neumann Algebra
Siberian Mathematical Journal, 2018Co-Authors: A M BikchentaevAbstract:Suppose that M is a Von Neumann Algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the Algebra M. The results are new even for the *-Algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.
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two classes of τ measurable operators affiliated with a Von Neumann Algebra
Russian Mathematics, 2017Co-Authors: A M BikchentaevAbstract: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 P 1 and P 2 of τ-measurable operators and investigate their properties. The class P 2 contains P 1. If a τ-measurable operator T is hyponormal, then T lies in P 1; if an operator T lies in P k , then UTU* belongs to P k for all isometries U from M and k = 1, 2; if an operator T from P 1 admits the bounded inverse T −1, then T −1 lies in P 1. We establish some new inequalities for rearrangements of operators from P 1. If a τ-measurable operator T is hyponormal and T n 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 P 1 coincides with the set of all paranormal operators on H.
Esteban Andruchow - One of the best experts on this subject based on the ideXlab platform.
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nonpositively curved metric in the positive cone of a finite Von Neumann Algebra
arXiv: Differential Geometry, 2008Co-Authors: Esteban Andruchow, Gabriel LarotondaAbstract:In this paper we study the metric geometry of the space $\Sigma$ of positive invertible elements of a Von Neumann Algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $\tau$. The trace induces an incomplete Riemannian metric $ _a=\tau (ya^{-1}xa^{-1})$, and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (Algebraic) characterization of the geodesically convex submanifolds $M$ of $\Sigma$, and under suitable hypothesis we prove a factorization theorem for elements in the Algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained \textit{via} a nonlinear orthogonal projection $\Pi_M:\Sigma\to M$, a map which turns out to be contractive for the geodesic distance.
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metric geometry of partial isometries in a finite Von Neumann Algebra
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Esteban AndruchowAbstract:Abstract We study the geometry of the set I p = { v ∈ M : v ∗ v = p } of partial isometries of a finite Von Neumann Algebra M, with initial space p (p is a projection of the Algebra). This set is a C ∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that ( I p , d g ) is a complete metric space, where d g is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).
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nonpositively curved metric in the positive cone of a finite Von Neumann Algebra
Journal of The London Mathematical Society-second Series, 2006Co-Authors: Esteban Andruchow, Gabriel LarotondaAbstract:In this paper we study the metric geometry of the spaceof positive invertible elements of a Von Neumann Algebra A with a finite, normal and faithful tracial state �. The trace induces an incomplete Riemannian metric a= �(ya −1 xa −1 ), and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the n × n matrices when they are regarded as a symmetric space. For instance we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (Algebraic) characterization of the geodesically convex submanifolds M of �, and under suitable hypothesis we prove a factorization theorem for elements in the Algebra that resembles the Iwasawa decomposition for matrices. This factoriza- tion is obtained via a nonlinear orthogonal projectionM : � → M, a map which turns out to be contractive for the geodesic distance. 1
Karimbergen Kudaybergenov - One of the best experts on this subject based on the ideXlab platform.
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derivations on the Algebra of τ compact operators affiliated with a type i Von Neumann Algebra
Positivity, 2008Co-Authors: Sergio Albeverio, Shavkat Ayupov, Karimbergen KudaybergenovAbstract:Let M be a type I Von Neumann Algebra with the center Z, and a faithful normal semi-finite trace τ. Consider the Algebra L(M, τ) of all τ-measurable operators with respect to M and let S 0(M, τ) be the subAlgebra of τ-compact operators in L(M, τ). We prove that any Z-linear derivation of S 0(M, τ) is spatial and generated by an element from L(M, τ).
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derivations on the Algebra of tau compact operators affiliated with a type i Von Neumann Algebra
arXiv: Operator Algebras, 2007Co-Authors: Sergio Albeverio, Sh A Ayupov, Karimbergen KudaybergenovAbstract:Let $M$ be a type I Von Neumann Algebra with the center $Z,$ a faithful normal semi-finite trace $\tau.$ Let $L(M, \tau)$ be the Algebra of all $\tau$-measurable operators affiliated with $M$ and let $S_0(M, \tau)$ be the subAlgebra in $L(M, \tau)$ consisting of all operators $x$ such that given any $\epsilon>0$ there is a projection $p\in\mathcal{P}(M)$ with $\tau(p^{\perp})<\infty, xp\in M$ and $\|xp\|<\epsilon.$ We prove that any $Z$-linear derivation of $S_0(M, \tau)$ is spatial and generated by an element from $L(M, \tau).$
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derivations on the Algebra of measurable operators affiliated with a type i Von Neumann Algebra
arXiv: Operator Algebras, 2007Co-Authors: Sergio Albeverio, Sh A Ayupov, Karimbergen KudaybergenovAbstract:Let $M$ be a type I Von Neumann Algebra with the center $Z,$ and let $LS(M)$ be the Algebra of all locally measurable operators affiliated with $M.$ We prove that every $Z$-linear derivation on $LS(M)$ is inner. In particular all $Z$-linear derivations on the Algebras of measurable and respectively totally measurable operators are spatial and implemented by elements from $LS(M).$
Uffe Haagerup - One of the best experts on this subject based on the ideXlab platform.
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brown measures of unbounded operators affiliated with a finite Von Neumann Algebra
Mathematica Scandinavica, 2007Co-Authors: Uffe Haagerup, Hanne SchultzAbstract:In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite Von Neumann Algebra. Moreover, we compute the Brown measure of all unbounded $R$-diagonal operators in this class. As a particular case, we determine the Brown measure $z=xy^{-1}$, where $(x,y)$ is a circular system in the sense of Voiculescu, and we prove that for all $n\in \mathsf N$, $z^n\in L^p$ if and only if $0
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brown measures of unbounded operators affiliated with a finite Von Neumann Algebra
arXiv: Operator Algebras, 2006Co-Authors: Uffe Haagerup, Hanne SchultzAbstract:In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite Von Neumann Algebra. Moreover, we compute the Brown measure of all unbounded R-diagonal operators in this class. As a particular case, we determine the Brown measure of z=xy^{-1}, where (x,y) is a circular system in the sense of Voiculescu, and we prove that for all positive integers n, z^n is in L^p(M) iff 0
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random matrices free probability and the invariant subspace problem relative to a Von Neumann Algebra
arXiv: Operator Algebras, 2002Co-Authors: Uffe HaagerupAbstract:Random matrices have their roots in multivariate analysis in statistics, and since Wigner's pioneering work in 1955, they have been a very important tool in mathematical physics. In functional analysis, random matrices and random structures have in the last two decades been used to construct Banach spaces with surprising properties. After Voiculescu in 1990--1991 used random matrices to classification problems for Von Neumann Algebras, they have played a key role in Von Neumann Algebra theory. In this lecture we will discuss some new applications of random matrices to operator Algebra theory, namely applications to classification problems for $C^*$-Algebras and to the invariant subspace problem relative to a Von Neumann Algebra.
Hanne Schultz - One of the best experts on this subject based on the ideXlab platform.
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brown measures of unbounded operators affiliated with a finite Von Neumann Algebra
Mathematica Scandinavica, 2007Co-Authors: Uffe Haagerup, Hanne SchultzAbstract:In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite Von Neumann Algebra. Moreover, we compute the Brown measure of all unbounded $R$-diagonal operators in this class. As a particular case, we determine the Brown measure $z=xy^{-1}$, where $(x,y)$ is a circular system in the sense of Voiculescu, and we prove that for all $n\in \mathsf N$, $z^n\in L^p$ if and only if $0
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brown measures of unbounded operators affiliated with a finite Von Neumann Algebra
arXiv: Operator Algebras, 2006Co-Authors: Uffe Haagerup, Hanne SchultzAbstract:In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite Von Neumann Algebra. Moreover, we compute the Brown measure of all unbounded R-diagonal operators in this class. As a particular case, we determine the Brown measure of z=xy^{-1}, where (x,y) is a circular system in the sense of Voiculescu, and we prove that for all positive integers n, z^n is in L^p(M) iff 0
