Derivations in the Banach ideals of τ\tau-compact operators

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $\tau$ and let $S_0(\tau)$ be the algebra of all $\tau$-compact operators affiliated with $\mathcal{M}$. Let $E(\tau)\subseteq S_0(\tau)$ be a symmetric operator space (on $\mathcal{M}$) and let $\mathcal{E}$ be a symmetrically-normed Banach ideal of $\tau$-compact operators in $\mathcal{M}$. We study (i) derivations $\delta$ on $\mathcal{M}$ with the range in $E(\tau)$ and (ii) derivations on the Banach algebra $\mathcal{E}$. In the first case our main results assert that such derivations are continuous (with respect to the norm topologies) and also inner (under some mild assumptions on $E(\tau)$). In the second case we show that any such derivation is necessarily inner when $\mathcal{M}$ is a type $I$ factor. As an interesting application of our results for the case (i) we deduce that any derivation from $\mathcal{M}$ into an $L_p$-space, $L_p(\mathcal{M},\tau)$, ($1<p<\infty$) associated with $\mathcal{M}$ is inner

Commutator estimates in W∗W^*-factors

Let $\mathcal{M}$ be a $W^*$-factor and let $S\left( \mathcal{M} \right)$ be the space of all measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in S(\mathcal{M})$ there exists a scalar $\lambda_0\in\mathbb{R}$, such that for all $\varepsilon > 0$, there exists a unitary element $u_\varepsilon$ from $\mathcal{M}$, satisfying $|[a,u_\varepsilon]| \geq (1-\varepsilon)|a-\lambda_0\mathbf{1}|$. A corollary of this result is that for any derivation $\delta$ on $\mathcal{M}$ with the range in an ideal $I\subseteq\mathcal{M}$, the derivation $\delta$ is inner, that is $\delta(\cdot)=\delta_a(\cdot)=[a,\cdot]$, and $a\in I$. Similar results are also obtained for inner derivations on $S(\mathcal{M})$.Comment: 21 page

Characteristics of enzymatic induction provoked by chlordane

The effects of various stresses, such as restraint and lowering or raising of environmental temperature, in mice pretreated with chlordane were investigated. (Chlordane is an inhibitor of protein synthesis.) It was found that restraint or exposure to a cold environment for three hours mobilized the chlordane stored in the adipose tissue of mice

Derivations on symmetric quasi-Banach ideals of compact operators

Let $\mathcal{I,J}$ be symmetric quasi-Banach ideals of compact operators on an infinite-dimensional complex Hilbert space $H$, let $\mathcal{J:I}$ be a space of multipliers from $\mathcal{I}$ to $\mathcal{J}$. Obviously, ideals $\mathcal{I}$ and $\mathcal{J}$ are quasi-Banach algebras and it is clear that ideal $\mathcal{J}$ is a bimodule for $\mathcal{I}$. We study the set of all derivations from $\mathcal{I}$ into $\mathcal{J}$. We show that any such derivation is automatically continuous and there exists an operator $a\in\mathcal{J:I}$ such that $\delta(\cdot)=[a,\cdot]$, moreover $\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to J}\leq 2C\|a\|_\mathcal{J:I}$, where $C$ is the modulus of concavity of the quasi-norm $\|\cdot\|_\mathcal{J}$. In the special case, when $\mathcal{I=J=K}(H)$ is a symmetric Banach ideal of compact operators on $H$ our result yields the classical fact that any derivation $\delta$ on $\mathcal{K}(H)$ may be written as $\delta(\cdot)=[a,\cdot]$, where $a$ is some bounded operator on $H$ and $\|a\|_{\mathcal{B}(H)}\leq\|\delta\|_\mathcal{I\to I}\leq 2\|a\|_{\mathcal{B}(H)}$.Comment: 21 page

Depth Profiling of Multilayer Mo/Si Nanostructures

A round-robin characterization is reported on the sputter depth profiling of [60(3.0 nm Mo/ 0.3 nm B4C/ 3.7 nm Si)] and [60 (3.5 nm Mo/ 3.5 nm Si)] stacks deposited on Si (111). Two different commercial secondary ion mass spectrometers with time-of-flight and magnetic-sector analyzers and a pulsed radio frequency glow discharge optical emission spectrometer were used. The pros and cons of each instrumental approach are discussed. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3526