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1. CJM Online first
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)
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)
Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators |
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)
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 |