Tensor products of subspace lattices and rank one density

We show that, if M is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, L is a commutative subspace lattice and P is the lattice of all projections on a separable Hilbert space, then L⊗M⊗P is reflexive. If M is moreover an atomic Boolean subspace lattice while L is any subspace lattice, we provide a concrete lattice theoretic description of L⊗M in terms of projection valued functions defined on the set of atoms of M . As a consequence, we show that the Lattice Tensor Product Formula holds for AlgM and any other reflexive operator algebra and give several further corollaries of these results.

Norm attaining operators and pseudospectrum

It is shown that if $11$, the operator $I+T$ attains its norm. A reflexive Banach space $X$ and a bounded rank one operator $T$ on $X$ are constructed such that $\|I+T\|>1$ and $I+T$ does not attain its norm.

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C0(X) into C0(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Snd ?hn for a (possibly finite) sequence {xn }n of distinct points in X and a norm null sequence {hn }n of mutually disjoint functions in C0(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Press perturbations and indirect effects in real food webs

The prediction of the effects of disturbances in natural systems is limited by the general lack of knowledge on the strength of species interactions, i.e., the effect of one species on the population growth rate of another, and by the uncertainty of the effects that may be manifested via indirect pathways within the food web. Here we explored the consequences of changes in species populations for the remaining species within nine exceptionally well-characterized empirical food webs, for which, unlike the vast majority of other published webs, feeding links have been fully quantied. Using the inverse of the Jacobian matrix, we found that perturbations to species with few connections have larger net effects (considering both direct and indirect pathways between two species) on the rest of the food web than do disturbances to species that are highly connected. For 40% of predator-prey links, predators had positive net effects on prey populations, due to the predominance of indirect interactions. Our results highlight the fundamental, but often counterintuitive, role of indirect effects for the maintenance of food web complexity and biodiversity.

Exploring Scale Effects of Best/Worst Rank Ordered Choice Data to Estimate Benefits of Tourism in Alpine Grazing Commons

In many environmental valuation applications standard sample sizes for choice modelling surveys are impractical to achieve. One can improve data quality using more in-depth surveys administered to fewer respondents. We report on a study using high quality rank-ordered data elicited with the best-worst approach. The resulting "exploded logit" choice model, estimated on 64 responses per person, was used to study the willingness to pay for external benefits by visitors for policies which maintain the cultural heritage of alpine grazing commons. We find evidence supporting this approach and reasonable estimates of mean WTP, which appear theoretically valid and policy informative. © The Author (2011).