Subadditivity Inequalities in von Neumann Algebras and Characterization of Tracial Functionals

We examine under which assumptions on a positive normal functional φ on a von Neumann algebra, and a Borel measurable function f: R ⁺ → R with f(0) = 0 the subadditivity inequality φ (f(A+B)) ≤ φ(f(A))+φ (f (B)) holds true for all positive operators A, B in . A corresponding characterization of tracial functionals among positive normal functionals on a von Neumann algebra is presented. O.E. Tikhonov - Supported by the Russian Foundation for Basic Research (grant no. 01-01-00129) and the scientific program Universities of Russia – Basic Research (grant no. UR.04.01.061).     

冯·诺依曼代数的可测算子的性质

本文研究了冯·诺依曼代数的可测算子的基本性质,定义了阶梯算子,证明了任意一个正可测算子可以由阶梯算子在定义域内按照强算子拓扑逼近,从而证明了任意一个可测算子可以由投影在定义域内按照强算子拓扑逼近.此外,还讨论了可测算子与有界算子的复合算子的可测性.     

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H₀, and let {T_t, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {T_t} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans [3] have shown that Lhas the form [formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H₀ K is a bounded operator satisfying the ‘minimality’condition that the set {(RX–(X)R)u, uH₀, XA} is totalin K, and K₀ is a fixed element of A. The unitality of {T_t}implies that L(1) = 0, and consequently K₀=iH–R^*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensen–Evans (CE) parameters which determine theCE generator L of the semigroup {T_t}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25.     

Operator Space Embedding of Schatten p-Classes Into Von Neumann Algebra Preduals

Let X₁ and X₂ be subspaces of quotients of R OH and C OH respectively. We use new free probability techniques to construct a completely isomorphic embedding of the Haagerup tensor product into the predual of a sufficiently large QWEP von Neumann algebra. As an immediate application, given any 1 < q ≤ 2, our result produces a completely isomorphic embedding of (equipped with its natural operator space structure) into with a QWEP von Neumann algebra. Received: June 2006, Revision: June 2007, Accepted: September 2007     

Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra

In this paper we study the metric geometry of the space ofpositive invertible elements of a von Neumann algebra A witha finite, normal and faithful tracial state . The trace inducesan incomplete Riemannian metric x,y_a = (ya^–1xa^–1),and, though the techniques involved are quite different, thesituation here resembles in many relevant aspects that of then x n matrices when they are regarded as a symmetric space.For instance, we prove that geodesics are the shortest pathsfor the metric induced, and that the geodesic distance is aconvex function; we give an intrinsic (algebraic) characterizationof the geodesically convex submanifolds M of ; and under a suitablehypothesis we prove a factorization theorem for elements inthe algebra that resembles the Iwasawa decomposition for matrices.This factorization is obtained via a nonlinear orthogonal projection_M : M, a map which turns out to be contractive for the geodesicdistance.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

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.     

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II₁ factors and Mn(C)) and symmetric gauge norms on L^∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II₁ factor coincides with the class of symmetric gauge norms on L^∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286-300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M₂(C)) and some extreme points of N(Mn(C)) (n?3). For a type II₁ factor M, we prove that if t (0?t?1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).     

Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

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).      

半局部环的von Neumann正则性

主要借助small-内射模(环),给出半局部环的正则性和强正则性的一些等价刻画.     

Von Neumann代数中的套子代数

本文主要讨论因子Ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上的线性满等距和自伴导子．证明了因子Ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上的每个线性满等距是同构乘酉算子或者是反同构乘酉算子；给出了其上自伴导子是内导子的条件并得到有限因子 Ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上的每个自伴导子都是内导子．     

On Fourier Orthogonal Projections in the Rotation Algebra

Projections are constructed in the rotation algebra that areorthogonal to their Fourier transform and which are fixed underthe flip automorphism. Such projections are expected in a constructionof an inductive limit structure for the irrational rotationalgebra that is invariant under the Fourier transform (namely,as two circle algebras of the same dimension, which are swappedby the Fourier transform, plus a few points). The calculationis based on Rieffel's construction of the Schwartz space asan equivalence bimodule of rotation algebras as well as on thetheory of theta functions.     

Variations of Constrained Domain Functionals Associated with Boundary-Value Problems

We study variational formulas for maximizers for domain functionalsF(x0, u(x0)), x0, and F(x,u(x))dxover all Lipschitz domains satisfying the constraintg(x) dx=1. Here, u is the solution ofa diffusion equation in . Functional variations arecomputed using domain variations which preserve the constraint exactly. Weshow that any maximizer solves a moving boundary problem for the diffusionequation. Further, we show that, for problems with symmetry, the optimaldomains are balls.     

Von Neumann J-algebras in a space with two symmetries

We show that a von Neumann J-algebra A of type (B) does not contain J-positive (J-negative) operators. J-projections in A are characterized. The class of plus-operators that are simultaneously self-adjoint and J-self-adjoint is described.     

Almost-Everywhere Convergence and (o)-Convergence in Rings of Measurable Operators Associated with a Finite von Neumann Algebra

We study the relationship between (o)-convergence and almost-everywhere convergence in the Hermite part of the ring of unbounded measurable operators associated with a finite von Neumann algebra. In particular, we prove a theorem according to which (o)-convergence and almost-everywhere convergence are equivalent if and only if the von Neumann algebra is of the type I.     

Banach Algebras Satisfying the Non-Unital Von Neumann Inequality

There is a Banach algebra satisfying the von Neumann inequalityfor polynomials in a single variable, without constant term,which is not isomorphic to a norm-closed algebra of operatorson a Hilbert space.     

Metrics with Positive Scalar Curvature at Infinity and Localization Algebra

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.     

On amenable groups of automorphisms on Von Neumann algebras

LetG ⊂ Aut ℳ be a countable group, ℳ a Von Neumann algebra. LetE be a set of pure states on ℳ such thatG*E ⊂E, S ^G be the set ofG invariant states on ℳ andS _E ^G =S ^G ∩w* cl coE. We investigate in this paper some geometric properties for the setS _E ^G which turn out to be equivalent to amenability for the groupG. For example, we show thatS _E ^G ⊂ ℳ_* (S _E ^G has the WRNP) implies that ℳ contains minimal projections (ê containsfinite G invariant orbits) hold true, for all ℳ iffG is amenable. Furthermore we show that ifG is amenable thenS ^G ∩M _* ^⊥ contains a big set, thus improving results obtained by Ching Chou in [2]. These results imply that no action of an amenable countable groupG on an arbitraryW* algebra ℳ iss — strongly ergodic. Moreover cardS ^G ∩M _* ^⊥ ≧2 ^c (see M. Choda [4], K. Schmidt [21] and compare with A. Connes and B. Weiss [5]). The author gratefully acknowledges the support of an Izaak Walton Killam Memorial Senior Fellowship.     

相应于一类对称自对偶李代数的顶点代数结构

设gsp是给定的对称自对偶李代数,给出了相应于gsp的一个顶点代数结构.     

C. Neumann System Associated with the Levi Hierarchy

51.Intr0ductionUsuallyasolitonequationcanbeexpressedasthecompatibleconditionofanover-determinedsystem,theso-calledLaxpair-Wehavea1readyknownthatthetwol1nearequationsintheLaxpaircanben0nlinearizedtobetw0c0mpatible,comp1etelyintegrableHamilt0niansystemsintheLioll"i1lesenseundersomecertainc0nstraintconditionsbetweentheeigenfunctionsandthepotentials(seeL1J).Am0ngtheseconstraintconditionsaretheretheBargmannconstraint{see[1,2,4j),theC.Neumannconstraint(see[l,4J),andthesymmetryconstra1nt(see[5,6j…