When each continuous operator is regular, II

The following theorem is essentially due to L.~Kantorovich and B. Vulikh and it describes one of the most important classes of Banach lattices between which each continuous operator is regular. {\bf Theorem 1.1.} {\sl Let $E$ be an arbitrary L-space and $F$ be an arbitrary Banach lattice with Levi norm. Then ${\cal L}(E,F)={\cal L}^r(E,F),\ (\star)$ that is, every continuous operator from $E$ to $F$ is regular.} In spite of the importance of this theorem it has not yet been determined to what extent the Levi condition is essential for the validity of equality $(\star)$. Our main aim in this work is to prove a converse to this theorem by showing that for a Dedekind complete $F$ the Levi condition is necessary for the validity of $(\star)$. As a sample of other results we mention the following. {\bf Theorem~3.6.} {\sl For a Banach lattice $F$ the following are equivalent: {\rm (a)} $F$ is Dedekind complete; {\rm (b)} For all Banach lattices $E$, the space ${\cal L}^r(E,F)$ is a Dedekind complete vector lattice; {\rm (c)} For all L-spaces $E$, the space ${\cal L}^r(E,F)$ is a vector lattice.

Whither Water Regulation?

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.Accepted author manuscriptAnalysi

Unique apicomplexan IMC sub-compartment proteins are early markers for apical polarity in the malaria parasite

The phylum Apicomplexa comprises over 5000 intracellular protozoan parasites, including Plasmodium and Toxoplasma, that are clinically important pathogens affecting humans and livestock. Malaria parasites belonging to the genus Plasmodium possess a pellicle comprised of a plasmalemma and inner membrane complex (IMC), which is implicated in parasite motility and invasion. Using live cell imaging and reverse genetics in the rodent malaria model P. berghei, we localise two unique IMC sub-compartment proteins (ISPs) and examine their role in defining apical polarity during zygote (ookinete) development. We show that these proteins localise to the anterior apical end of the parasite where IMC organisation is initiated, and are expressed at all developmental stages, especially those that are invasive. Both ISP proteins are N-myristoylated, phosphorylated and membrane-bound. Gene disruption studies suggest that ISP1 is likely essential for parasite development, whereas ISP3 is not. However, an absence of ISP3 alters the apical localisation of ISP1 in all invasive stages including ookinetes and sporozoites, suggesting a coordinated function for these proteins in the organisation of apical polarity in the parasite