On Wolfgang Lusky's paper "The Gurarij spaces are unique''
This note surveys Wolfgang Lusky's proof of uniqueness of the Gurariy spaces and mentions further developments.
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arXiv subjects
Dirk Werner
Publications and source records attributed to Dirk Werner.
The Perfekt theory of $M$-ideals
We revisit some ideas of K.-M.~Perfekt who has provided an elegant framework to detect the biduality between function or sequence spaces defined in terms of some $o$- resp.\ $O$-condition. We present new proofs under somewhat weaker assumptions than before and apply the result to Lipschitz spaces.
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Norm attaining operators of finite rank
We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a non-trivial cone consisting of norm attaining functionals on $X$. We go on to discuss density of norm attaining operators of finite rank among all operators of finite rank, which holds for instance when there is a dense linear subspace consisting of norm attaining functionals on $X$. In particular, we consider the case of Hilbert space valued operators where we obtain a complete characterization of these properties. In the final section we offer a candidate for a counterexample to the complex Bishop-Phelps theorem on $c_0$, the first such counterexample on a certain complex Banach space being due to V. Lomonosov.
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Totally smooth renormings
We study the problem of totally smooth renormings of Banach spaces and provide such renormings for spaces which are weakly compactly generated. We also consider renormings for $(a,B,c)$-ideals.
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Operations with slicely countably determined sets
The notion of slicely countably determined (SCD) sets was introduced in 2010 by A.~Avilés, V.~Kadets, M.~Martín, J.~Merí and V.~Shepelska. We solve in the negative some natural questions about preserving being SCD by the operations of union, intersection and Minkowski sum. Moreover, we demonstrate that corresponding examples exist in every space with the Daugavet property and can be selected to be unit balls of some equivalent norms. We also demonstrate that almost SCD sets need not be SCD, thus answering a question posed by A. Avilés et al.
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Equivalent norms with an extremely nonlineable set of norm attaining functionals
We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read and Rmoutil. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $NA(X)$ for the original norm is not "too large". The construction can be applied to every Banach space containing $c_0$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_0$. We also provide some geometric properties of the norms we have constructed.
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Approximation of integration maps of vector measures and limit representations of Banach function spaces
We study when the integration maps of vector measures can be computed as pointwise limits of their finite rank Radon-Nikodým derivatives. We will show that this can sometimes be done, but there are also principal cases in which this cannot be done. The positive cases are obtained using the circle of ideas of the approximation property for Banach spaces. The negative ones are given by means of an adequate use of the Daugavet property. As an application, we analyse when the norm in a space of integrable functions $L^1(m)$ can be computed as a limit of the norms of the spaces of integrable functions with respect to the Radon-Nikodým derivatives of $m$.
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The Daugavet equation for bounded vector valued functions
Requirements under which the Daugavet equation and the alternative Daugavet equation hold for pairs of nonlinear maps between Banach spaces are analysed. A geometric description is given in terms of nonlinear slices. Some local versions of these properties are also introduced and studied, as well as tests for checking if the required conditions are satisfied in relevant cases.
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Lipschitz slices and the Daugavet equation for Lipschitz operators
We introduce a substitute for the concept of slice for the case of non-linear Lipschitz functionals and transfer to the non-linear case some results about the Daugavet and the alternative Daugavet equations previously known only for linear operators.
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Slice continuity for operators and the Daugavet property for bilinear maps
We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced.
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Nigel Kalton's work in isometrical Banach space theory
This paper surveys some of the late Nigel Kalton's contributions to Banach space theory. The paper is written for the Nigel Kalton Memorial Website http://mathematics.missouri.edu/kalton/, which is scheduled to go online in summer 2011.
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Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces
We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index~1 are equivalent. In the class of rearrangement invariant (r.i.)\ sequence spaces the only examples of spaces with these properties are $c_0$, $\ell_1$ and $\ell_\infty$. The only lush r.i.\ separable function space on $[0,1]$ is $L_1[0,1]$; the same space is the only r.i.\ separable function space on $[0,1]$ with the Daugavet property over the reals.
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The $p$-Daugavet property for function spaces
A natural extension of the Daugavet property for $p$-convex Banach function spaces and related classes is analysed. As an application, we extend the arguments given in the setting of the Daugavet property to show that no reflexive space falls into this class.
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The geometry of L^p-spaces over atomless measure spaces and the Daugavet property
We show that $L^p$-spaces over atomless measure spaces can be characterized in terms of a $p$-concavity type geometric property that is related with the Daugavet property.
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Thickness of the unit sphere, $\ell_1$-types, and the almost Daugavet property
We study those Banach spaces $X$ for which $S_X$ does not admit a finite $\eps$-net consisting of elements of $S_X$ for any $\eps < 2$. We give characterisations of this class of spaces in terms of $\ell_1$-type sequences and in terms of the almost Daugavet property. The main result of the paper is: a separable Banach space $X$ is isomorphic to a space from this class if and only if $X$ contains an isomorphic copy of $\ell_1$.
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Quotients of Banach spaces with the Daugavet property
We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative.
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The Daugavet property for spaces of Lipschitz functions
For a compact metric space $K$ the space $\Lip(K)$ has the Daugavet property if and only if the norm of every $f \in \Lip(K)$ is attained locally. If $K$ is a subset of an $L_p$-space, $1<p<\infty$, this is equivalent to the convexity of $K$.
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Numerical index and duality
We present an example of a Banach space whose numerical index is strictly greater than the numerical index of its dual, giving a negative answer to a question which has been latent since the beginning of the seventies. We also show a particular case in which the numerical index of the space and the one of its dual coincide.
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