Multidimensional operator multipliers

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C *-algebras satisfying certain boundedness conditions. In the case of commutative C*-algebras, the multidimensional operator multipliersreduce to continuousmul-tidimensional Schur multipliers. We show that the multiplierswith respect to some given representations of the corresponding C*-algebrasdo not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained ascertain weak limits of elements of the algebraic tensor product of the corresponding C *-algebras.

Compactness properties of operator multipliers

We continue the study of multidimensional operator multipliers initiated in~cite{jtt}. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C*-algebra of compact operators in terms of tensor products, generalising results of Saar

Electrostatic mode envelope excitations in e-p-i plasmas - application in warm pair ion plasmas with a small fraction of stationary ions

The nonlinear propagation of amplitude-modulated electrostatic wavepackets in an electron-positron-ion (e-p-i) plasma is considered, by employing a two-fluid plasma model. Considering propagation parallel to the external magnetic field, two distinct electrostatic modes are obtained, namely a quasi-thermal acoustic-like lower mode and a Langmuir-like optic-type upper one. These results equally apply in warm pair ion ( e. g. fullerene) plasmas contaminated by a small fraction of stationary ions ( or dust), in agreement with experimental observations and theoretical predictions in pair plasmas. Considering small yet weakly nonlinear deviations from equilibrium, and adopting a multiple-scales perturbation technique, the basic set of model equations is reduced to a nonlinear Schrodinger (NLS) equation for the slowly varying electric field perturbation amplitude. The analysis reveals that the lower ( acoustic) mode is mostly stable for large wavelengths, and may propagate in the form of a dark-type envelope soliton ( a void) modulating a carrier wavepacket, while the upper linear mode is intrinsically unstable, and thus favours the formation of bright-type envelope soliton ( pulse) modulated wavepackets. The stability ( instability) range for the acoustic ( Langmuir-like optic) mode shifts to larger wavenumbers as the positive-to-negative ion temperature ( density) ratio increases. These results may be of relevance in astrophysical contexts, where e-p-i plasmas are encountered, and may also serve as prediction of the behaviour of doped ( or dust-contaminated) fullerene plasmas, in the laboratory.

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.

Multipliers of multidimensional Fourier algebras

Let $G$ be a locally compact $\sigma$-compact group. Motivated by an earlier notion for discrete groups due to Effros and Ruan, we introduce the multidimensional Fourier algebra $A^n(G)$ of $G$. We characterise the completely bounded multidimensional multipliers associated with $A^n(G)$ in several equivalent ways. In particular, we establish a completely isometric embedding of the space of all $n$-dimensional completely bounded multipliers into the space of all Schur multipliers on $G^{n+1}$ with respect to the (left) Haar measure. We show that in the case $G$ is amenable the space of completely bounded multidimensional multipliers coincides with the multidimensional Fourier-Stieltjes algebra of $G$ introduced by Ylinen. We extend some well-known results for abelian groups to the multidimensional setting.

Construction of Drug–Polymer Thermodynamic Phase Diagrams Using Flory–Huggins Interaction Theory: Identifying the Relevance of Temperature and Drug Weight Fraction to Phase Separation within Solid Dispersions

Amorphous drug-polymer solid dispersions have the potential to enhance the dissolution performance and thus bioavailability of BCS class II drug compounds. The principle drawback of this approach is the limited physical stability of amorphous drug within the dispersion. Accurate determination of the solubility and miscibility of drug in the polymer matrix is the key to the successful design and development of such systems. In this paper, we propose a novel method, based on Flory-Huggins theory, to predict and compare the solubility and miscibility of drug in polymeric systems. The systems chosen for this study are (1) hydroxypropyl methylcellulose acetate succinate HF grade (HPMCAS-HF)-felodipine (FD) and (2) Soluplus (a graft copolymer of polyvinyl caprolactam-polyvinyl acetate-polyethylene glycol)-FD. Samples containing different drug compositions were mixed, ball milled, and then analyzed by differential scanning calorimetry (DSC). The value of the drug-polymer interaction parameter ? was calculated from the crystalline drug melting depression data and extrapolated to lower temperatures. The interaction parameter ? was also calculated at 25 °C for both systems using the van Krevelen solubility parameter method. The rank order of interaction parameters of the two systems obtained at this temperature was comparable. Diagrams of drug-polymer temperature-composition and free energy of mixing (?G mix) were constructed for both systems. The maximum crystalline drug solubility and amorphous drug miscibility may be predicted based on the phase diagrams. Hyper-DSC was used to assess the validity of constructed phase diagrams by annealing solid dispersions at specific drug loadings. Three different samples for each polymer were selected to represent different regions within the phase diagram

The effect of multiple micronutrient supplementation on left ventricular ejection fraction in patients with chronic stable heart failure: a randomized, placebo-controlled trial:A randomized, placebo-controlled trial

Objectives: This study sought to investigate the effect of a multiple micronutrient supplement on left ventricular ejection fraction (LVEF) in patients with heart failure. Background: Observational studies suggest that patients with heart failure have reduced intake and lower concentrations of a number of micronutrients. However, there have been very few intervention studies investigating the effect of micronutrient supplementation in patients with heart failure. Methods: This was a randomized, double-blind, placebo-controlled, parallel-group study involving 74 patients with chronic stable heart failure that compared multiple micronutrient supplementation taken once daily versus placebo for 12 months. The primary endpoint was LVEF assessed by cardiovascular magnetic resonance imaging or 3-dimensional echocardiography. Secondary endpoints were Minnesota Living With Heart Failure Questionnaire score, 6-min walk test distance, blood concentrations of N-terminal prohormone of brain natriuretic peptide, C-reactive protein, tumor necrosis factor alpha, interleukin-6, interleukin-10, and urinary levels of 8-iso-prostaglandin F2 alpha. Results: Blood concentrations of a number of micronutrients increased significantly in the micronutrient supplement group, indicating excellent compliance with the intervention. There was no significant difference in mean LVEF at 12 months between treatment groups after adjusting for baseline (mean difference: 1.6%, 95% confidence interval: -2.6 to 5.8, p = 0.441). There was also no significant difference in any of the secondary endpoints at 12 months between treatment groups. Conclusions: This study provides no evidence to support the routine treatment of patients with chronic stable heart failure with a multiple micronutrient supplement. (Micronutrient Supplementation in Patients With Heart Failure [MINT-HF]; NCT01005303).

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).

Local operator multipliers and positivity

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and characterise them extending both the characterisation of operator multipliers from [16] and that of local Schur multipliers from [27]. We provide a description of the positive local operator multipliers in terms of approximation by elements of canonical positive cones.