Corrigendum to the paper ``Adjoining an Order Unit to a Matrix Ordered Space'

An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI ). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.     

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).      

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

Approximation properties and absence of Cartan subalgebra for free Araki–Woods factors

We show that all the free Araki–Woods factors Γ^″(HR,Ut) have the complete metric approximation property. Using Ozawa–Popa?s techniques, we then prove that every nonamenable subfactor N⊂Γ^″(HR,Ut) which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type III₁ factors constructed by Connes in the ?70s can never be isomorphic to any free Araki–Woods factor, which answers a question of Shlyakhtenko and Vaes.     

Characterizations of Noncommutative H ∞

We transfer a large part of the circle of theorems characterizing the generalization of classical H^∞ known as ‘weak* Dirichlet algebras’, to Arveson’s very general noncommutative setting of subalgebras of finite von Neumann algebras. This solves the long-standing open question of the equivalence of principles such as Szeg?’s theorem, the weak* density of A + A*, and so on, within the noncommutative setting. The techniques should also be useful in future developments in noncommutative H^p theory.     

On ergodic theorems for free group actions on noncommutative spaces

We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s_2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s_2n(x)) for x in noncommutative spaces L^p(A). For the Cesàro means this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende Verfahren' theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.     

Stability of Ternary Homomorphisms via Generalized Jensen Equation

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of homomorphisms between ternary algebras associated to the generalized Jensen functional equation .     

Boundary Behavior of Functions in the de Branges–Rovnyak Spaces

This paper deals with the boundary behavior of functions in the de Branges–Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges–Rovnyak spaces. This criterion generalizes a result of Ahern–Clark. Then we prove that the continuity of all functions in a de Branges–Rovnyak space on an open arc I of the boundary is enough to ensure the analyticity of these functions on I. We use this property in a question related to Bernstein’s inequality. Received: May 10, 2007. Revised: August 8, 2007. Accepted: August 8, 2007.     

Spectra for solvable Lie algebras of bundle endomorphisms

The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of -algebras. The flag manifolds in -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles. Received: 8 October 2001 / Revised version: 4 February 2002 / Published online: 6 August 2002 Research supported from the contract ICA1–CT–2000–70022 with the European Commission.     

Positive Projective Space of a <Emphasis Type="Italic">C</Emphasis>*-Algebra with a Tracial Conditional Expectation

Let be a C*-algebra, a subalgebra of its center and Φ: → a tracial faithful conditional expectation. We define the positive projective space as the quotient where G⁺ is the space of positive invertible elements of , and if there exists g invertible in such that a′ = |g|²a. When is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow ℙ⁺ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in ℙ⁺, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space ℙ⁺ with the given geodesic distance is non positively curved.     

The Radon–Nikodym Property for Tensor Products of Banach Lattices

Let 1 ≤ p < ∞. We show that , the Fremlin projective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that , the Wittstock injective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from ℓ_p' to X is compact, where 1/p +1/p'= 1 and let ℓ_p' = c₀ if p = 1. The author gratefully acknowledges support from the Office of Naval Research Grant # N00014-03-1-0621     

From Brunn–Minkowski to sharp Sobolev inequalities

We present a simple direct proof of the classical Sobolev inequality in with best constant from the geometric Brunn–Minkowski–Lusternik inequality. Research supported in part by NSF Gr. No. 0405587 and B. Zegarlinski’s Pierre de Fermat Grant 2006 from the Région Midi-Pyrénées, France.     

Balls have the worst best Sobolev inequalities. Part II: variants and extensions

We continue our previous study of sharp Sobolev-type inequalities by means of optimal transport, started in (Maggi and Villani J. Geom. Anal. 15(1), 83–121 (2005)). In the present paper, we extend our results in various directions, including Gagliardo–Nirenberg, Faber–Krahn, logarithmic-Sobolev or Moser–Trudinger inequalities with trace terms. We also identify a class of domains for which there is no need for a trace term to cast the Sobolev inequality.     

The non-commutative Gurarii space

We construct a “universal space” for 1-exact finite dimensional operator spaces (an analogue of the Gurarii space). Received: 20 June 2005     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033     

Visualization and art in the mathematics classroom

In this paper we summarize our concepts and practice on computer-aided mathematical experimentation, and illustrate them byMathematica projects that we have developed for our research and the courses “Computer-aided mathematical modelling” and “Computer Algebra I–II” held for students of life sciences at University of Szeged and computational engineering at TFH Berlin, University of Applied Sciences.     

Decomposition Methods in Banach Spaces via Supplemented Subspaces Resembling Pełczyński’s Decomposition Methods

Suppose that X and Y are Banach spaces complemented in each other with supplemented subspaces A and B. In 1996, W. T. Gowers solved the Schroeder–Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain some suitable conditions involving the spaces A and B to yield that X is isomorphic to Y or to provide that at least X ^m is isomorphic to Yⁿ for some m, n ∈ IN^*. So we get some decomposition methods in Banach spaces via supplemented subspaces resembling Pełczyński’s decomposition methods. In order to do this, we introduce several notions of Schroeder–Bernstein Quadruples acting on the spaces X, Y, A and B. Thus, we characterize them by using some Banach spaces recently constructed. Received: October 4, 2005.     

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001