Separable Banach lattices on which every bounded linear operator is regular

We give a complete description of those separable Banach lattices E with the property that every bounded linear from E into itself is the difference of two positive operators.

Closable multipliers

Let $(X,\mu)$ and $(Y,\nu)$ be standard measure spaces. A function $\nph\in L^\infty(X\times Y,\mu\times\nu)$ is called a (measurable) Schur multiplier if the map $S_\nph$, defined on the space of Hilbert-Schmidt operators from $L_2(X,\mu)$ to $L_2(Y,\nu)$ by multiplying their integral kernels by $\nph$, is bound-ed in the operator norm. The paper studies measurable functions $\nph$ for which $S_\nph$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if $\nph$ is of Toeplitz type, that is, if $\nph(x,y)=f(x-y)$, $x,y\in G$, where $G$ is a locally compact abelian group, then the closability of $\nph$ is related to the local inclusion of $f$ in the Fourier algebra $A(G)$ of $G$. If $\nph$ is a divided difference, that is, a function of the form $(f(x)-f(y))/(x-y)$, then its closability is related to the ``operator smoothness'' of the function $f$. A number of examples of non closable, norm closable and w*-closable multipliers are presented.

Infinite Dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on Rn failing the Denjoy-Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function.

On the spectrum of frequently hypercyclic operators

A bounded linear operator $T$ on a Banach space $X$ is called frequently hypercyclic if there exists $x\in X$ such that the lower density of the set $\{n\in\N:T^nx\in U\}$ is positive for any non-empty open subset $U$ of $X$. Bayart and Grivaux have raised a question whether there is a frequently hypercyclic operator on any separable infinite dimensional Banach space. We prove that the spectrum of a frequently hypercyclic operator has no isolated points. It follows that there are no frequently hypercyclic operators on all complex and on some real hereditarily indecomposable Banach spaces, which provides a negative answer to the above question.

1-amenability of A(X) for Banach spaces with 1-unconditional bases

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property A has in fact the property. Some further ideas on the problem of whether or not amenability (in this setting) implies property A are discussed.

Sensing Spaces:A Body-oriented Architectural Design Education

The contemporary dominance of visuality has turned our understanding of space into a mode of unidirectional experience that externalizes other sensual capacities of the body while perceiving the built environment. This affects not only architectural practice but also architectural education when an introduction to the concept of space is often challenging, especially for the students who have limited spatial and sensual training. Considering that an architectural work is not perceived as a series of retinal pictures but as a repeated multi-sensory experience, the problem definitions in the design studio need to be disengaged from the dominance of a ‘focused vision’ and be re-constructed in a holistic manner. A method to address this approach is to enable the students to refer to their own sensual experiences of the built environment as a part of their design processes. This paper focuses on a particular approach to the second year architectural design teaching which has been followed in the Department of Architecture at Izmir University of Economics for the last three years. The very first architectural project of the studio and the program, entitled ‘Sensing Spaces’, is conducted as a multi-staged design process including ‘sense games, analyses of organs and their interpretations into space’. The objectives of this four-week project are to explore the sense of space through the design of a three-dimensional assembly, to create an awareness of the significance of the senses in the design process and to experiment with re-interpreted forms of bodily parts. Hence, the students are encouraged to explore architectural space through their ‘tactile, olfactory, auditory, gustative and visual stimuli’. In this paper, based on a series of examples, architectural space is examined beyond its boundaries of structure, form and function, and spatial design is considered as an activity of re-constructing the built environment through the awareness of bodily senses.