Applications of the complex interpolation method to a von Neumann algebra: Non-commutative Lp-spaces

AbstractNon-commutative Lp-spaces, 1 < p < ∞, associated with a von Neumann algebra are considered. The paper consists of two parts. In part I, by making use of the complex interpolation method, non-commutative Lp-spaces are defined as interpolation spaces between the von Neumann algebra in question and its predual. Also, all expected properties (such as duality and uniform convexity) are proved in the frame of interpolaton theory and relative modular theory. In part II, these Lp-spaces are compared with Haagerup's Lp-spaces. Based on this comparison, a non-commutative analogue of the classical Stein-Weiss interpolation theorem is obtained

Families of completely positive maps associated with monotone metrics

An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps. We study the question of when these operators define maps which are also completely positive (CP). Although A --> D^{-1/2} A D^{-1/2} is the only case for which both the map and its inverse are CP, there are several well-known one parameter families for which either the map or its inverse is CP. We present a complete analysis of the behavior of these families, as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points. Our primary tool is an order relation based on the concept of positive definite functions. Although some results can be obtained from known properties, we also prove new results based on the positivity of the Fourier transforms of certain functions. Concrete computations of certain Fourier transforms not only yield new examples of positive definite functions, but also examples in the much stronger class of infinitely divisible functions.Comment: Final version to appear in Lin Alg. Appl. Links to preprints adde

On infinite divisibility of positive definite functions arising from operator means

AbstractFor a (point-wisely non-negative) positive definite function a certain criterion for its infinite divisibility (i.e., all its fractional powers are also positive definite) is obtained. This criterion enables us to show infinite divisibility for many positive definite functions appearing naturally in study of operator means. In particular, we determine when the functioncosh(νx)+s′coshx+s(ν∈[0,1];s,s′∈(−1,1]) is infinitely divisible

Environmental studies on the fate and effects of pharma proteins produced in transgenic maize

The environmental fate and effects of a transgenic maize-expressed recombinant antigen, Escherichia coli heat-labile enterotoxin subunit B (LTB), was studied under laboratory conditions. The time for 50% dissipation (DT50) for the extractable fraction of maize-expressed LTB in soil was 35∼90 days, and both extractability and persistence were strongly affected by the soil types and environmental conditions. In addition to the environmental fate study, laboratory bioassays were conducted to investigate the sub-acute effects of maize-expressed LTB on two soil invertebrates. Both the 28-day springtail, Folsomia candida reproduction assay and earthworm, Eisenia fetida growth assay showed no adverse effects of LTB under a concentration which was well above the maximum estimated environmental concentrations. For these observations, it is predicted that the adverse effects of LTB expressing maize on F. candida and E. fetida are negligible

A remark on the minimal index of subfactors

AbstractLet M ⫆ N be a factor-subfactor pair with a faithful normal conditional expectation E: M → N satisfying Ind E < + ∞. Let N ⫅ M ⫅ M1 ⫅ M2 ··· be the Jones tower of basic extensions, and Ek: Mk → Mk − 1 be the dual expectation constructed canonically from the given E. We prove that, if E: M → N is minimal, then so is the composition E ° E1 ° ··· ° Ek: Mk → N. Some consequences of this result are also presented