Narrow operators and the Daugavet property for ultraproducts

We show that if T is a narrow operator (for the definition see below) on or , then the restrictions to X₁ and X₂ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.     

The alternative Daugavet property of C *‐algebras and JB *‐triples

A Banach space X is said to have the alternative Daugavet property if for every (bounded and linear) rank‐one operator T: X → X there exists a modulus one scalar ω such that ∥Id+ωT ∥ = 1 + ∥T ∥. We give geometric characterizations of this property in the setting of C *‐algebras, JB *‐triples, and of their isometric preduals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An alternative Daugavet property

We introduce a strictly weaker version of the Daugavet property as follows: a Banach space X has this alternative Daugavet property (ADP in short) if the norm identity

(aDE)     

The polynomial Daugavet property for atomless L 1(μ)-spaces

For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial ${P:L_1(\mu)\longrightarrow L_1(\mu)}For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial P:L₁(m)? L₁(m){P:L_1(\mu)\longrightarrow L_1(\mu)} satisfies the Daugavet equation ||Id + P||=1 + ||P||{\|{\rm Id} + P\|=1 + \|P\|}. The same is true for the vector-valued spaces L ₁(μ, E), μ atomless, E arbitrary.     

Representable spaces have the polynomial Daugavet property

In this note we prove that every Banach space that is representable in a compact Hausdorff topological space in the sense of (JFunct Anal 254:2294–2302, 2008) has the polynomial Daugavet property. As an application we provide new examples of Banach spaces enjoying the polynomial Daugavet property.     

A Banach space with the Schur and the Daugavet property

We show that a minor refinement of the Bourgain-Rosenthal construction of a Banach space without the Radon-Nikodým property which contains no bounded -trees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such spaces; in particular, we show that the Daugavet property is not inherited by ultraproducts.

     

The Daugavet Property of C*-Algebras and Non-commutative Lp-Spaces

We prove that a C ^*-algebra A or a predual N _* of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative L _p-spaces corresponding to non-atomic von Neumann algebras.     

On the structure of spaces of uniformly convergent Fourier series

We first give some new examples of translation invariant subspaces of C or U without local unconditional structure. In the second part, we prove that U and U ⁺ do not have the Gordon–Lewis property. In the third part, we show that absolutely summing operators from U to a K-convex space are compact. As a consequence, U and U ⁺ are not isomorphic. At last, we prove that U and U ⁺ do not have the Daugavet property.     

Daugavet centers and direct sums of Banach spaces

A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X _1⊕F X ₂ of two Banach spaces X ₁ and X ₂ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X ₁ and X ₂ there exists a Daugavet center acting from X _1⊕F X ₂, and the class of those F such that for some pair of spaces X ₁ and X ₂ there is a Daugavet center acting into X _1⊕F X ₂. We also present several examples of such Daugavet centers.     

SOME REMARKS CONCERNING THE DAUGAVET EQUATION

Abstract

We say that a normed space X has the Daugavet property (DP) if for every finite rank operator K in X the equality ∥I + T∥ = 1 + ∥T∥ holds. It is known that C[0,1] and L ₁[0,1] have DP. We prove that if X has DP then X has no unconditional basis. We also discuss anti-Daugavet property, hereditary DP-spaces and construct a strictly convex normed space having DP.     

Sums of SCD sets and their applications to SCD operators and narrow operators

We answer two open questions concerning the recently introduced notions of slicely countably determined (SCD) sets and SCD operators in Banach spaces. An application to narrow operators in spaces with the Daugavet property is given.     

An exact Daugavet type inequality for small into isomorphisms in L 1

The well known Daugavet property for the space L ₁ means that || I  +  K || = 1+ || K || for any weakly compact operator K : L ₁ → L ₁, where I is the identity operator in L ₁. We generalize this theorem to the case when we consider an into isomorphism J : L ₁ → L ₁ instead of I and a narrow operator T. Our main result states that , where d  =  || J|| || J ⁻¹||. We also give an example which shows that this estimate is exact. Received: 21 August 2007     

On the geometry of von Neumann algebra preduals

Let \(M\) be a von Neumann algebra and let \(M_\star \) be its (unique) predual. We study when for every \(\varphi \in M_\star \) there exists \(\psi \in M_\star \) solving the equation \(\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert \) . This is the case when \(M\) does not contain type I nor type III \(_1\) factors as direct summands and it is false at least for the unique hyperfinite type III \(_1\) factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of \(M_\star \) of length \(4\) . An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.     

Octahedral norms and convex combination of slices in Banach spaces

We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of w^?$w^{?}$-slices in the dual unit ball has diameter 2, which answers an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing _?1${?}_1$ and for every ε>0$\epsilon >0$ there is an equivalent norm so that every convex combination of w^?$w^{?}$-slices in the dual unit ball has diameter at least 2−ε$2-\epsilon$.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

<Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript> transversality and shadowing properties

Let f be an Axiom A diffeomorphism of a closed smooth two-dimensional manifold. It is shown that the following statements are equivalent: (a) f satisfies the C ⁰ transversality condition, (b) f has the shadowing property, and (c) f has the inverse shadowing property with respect to a class of continuous methods.     

Diffeomorphisms with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-shadowing property

We give a characterization of structurally stable diffeomorphisms by making use of the notion of L ^p -shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C ¹-interior of the set of diffeomorphisms having L ^p -shadowing property.     

Simple <Emphasis Type="Italic">SL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-modules with normal closures of maximal torus orbits

Let T be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules V with the following property: for each point v∈V the closure [`(Tv)]\overline{Tv} of its T-orbit is a normal affine variety. Moreover, for any SL(n)-module without this property a T-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL(n)-modules. The saturation property is checked for each subset in the set of weights.     

On <Emphasis Type="Italic">CAS</Emphasis>-subgroups of finite groups

We introduce a new subgroup embedding property of a finite group called CAS-subgroup. Using this subgroup property, we determine the structure of finite groups with some CAS-subgroups of Sylow subgroups. Our results unify and generalize some recent theorems on solvability, p-nilpotency and supersolvability of finite groups. The authors are supported by NSF of China (10571181) and NSF of Guangxi (0447038).