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1. CJM 2013 (vol 66 pp. 1143)

Plevnik, Lucijan; Šemrl, Peter
Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space
Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi , \psi : {\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair $U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents
Categories:46B20, 47B49

2. CJM 2010 (vol 62 pp. 870)

Valdimarsson, Stefán Ingi
The Brascamp-Lieb Polyhedron
A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list.

Keywords:Brascamp-Lieb inequality, Loomis-Whitney inequality, lattice, flag
Categories:44A35, 14M15, 26D20

3. CJM 2007 (vol 59 pp. 614)

Labuschagne, C. C. A.
Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators
We use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of $p$-convex, $p$-concave and positive $p$-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Keywords:$p$-convex operator, $p$-concave operator, $p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence space
Categories:46B28, 47B10, 46B42, 46B45

4. CJM 1999 (vol 51 pp. 792)

Grätzer, G.; Wehrung, F.
Tensor Products and Transferability of Semilattices
In general, the tensor product, $A \otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a {\it capped\/} tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for {\it every\/} lattice $B$ with zero; we shall call such lattices {\it amenable}. The first author introduced in 1966 the concept of a {\it sharply transferable lattice}. In 1972, H.~Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that {\it a finite lattice $A$ is\/} amenable {\it if{}f it is\/} sharply transferable {\it as a join-semilattice}. For a general lattice $A$ with zero, we obtain the result: {\it $A$ is amenable if{}f $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice}. This yields, for example, that a finite lattice $A$ is amenable if{}f $A\otimes\FL(3)$ is a lattice if{}f $A$ satisfies (T), with respect to join. In particular, $M_3\otimes\FL(3)$ is not a lattice. This solves a problem raised by R.~W.~Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

Keywords:tensor product, semilattice, lattice, transferability, minimal pair, capped
Categories:06B05, 06B15

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