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تسجيل مستخدم جديدInequalities for trace on $tau$-measurable operators
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Let $mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $tau$. A closed densely defined operator $x$ affiliated with $mathfrak{M}$ is called $tau$-measurable if there exists a number $lambda geq 0$ such that $tau left(e^{|x|}(lambda,infty)right)<infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $tau$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $tau$-measurable operators. A general parallelogram law for $tau$-measurable operators are given as well.
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Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$)
be the algebra of all $tau$-measurable (resp. $tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $tau,$ let $S_0(M, tau)$ be the algebra of all $tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, tau).
$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $S_0(M, tau)$ is spatial.
We establish that trace inequalities $$|D^{k-1}u|_{L^{frac{n-s}{n-1}}(mathbb{R}^{n},dmu)} leq c |mu|_{L^{1,n-s}(mathbb{R}^{n})}^{frac{n-1}{n-s}}|mathbb{A}[D]u|_{L^{1}(mathbb{R}^{n},dmathscr{L}^{n})}$$ hold for vector fields $uin C^{infty}(mathbb{R}^{
n};mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $mathbb{A}[D]$ on $mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $mathbb{A}[D]$ are necessary. Here, $|mu|_{L^{1,lambda}}$ denotes the $(1,lambda)$-Morrey norm of the measure $mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($mathbb{A}$-)strict continuity of the associated trace operators on $text{BV}^{mathbb{A}}$.
The parallel sum for adjoinable operators on Hilbert $C^*$-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert $C^*$-modules
. It is shown that there exist a Hilbert $C^*$-module $H$ and two positive operators $A, Binmathcal{L}(H)$ such that the operator equation $A^{1/2}=(A+B)^{1/2}X, Xin cal{L}(H)$ has no solution, where $mathcal{L}(H)$ denotes the set of all adjointable operators on $H$.
We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner product struct
ure of a Hilbert $C^*$-module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving mappings are investigated, not a priori bounded. The intuition is that most often $C^*$-linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if $mathscr{A}$ is a $C^{*}$-algebra and $T, S:mathscr{E}longrightarrow mathscr{F}$ are two bounded ${mathscr A}$-linear mappings between full Hilbert $mathscr{A}$-modules, then $langle x, yrangle = 0$ implies $langle T(x), S(y)rangle = 0$ for all $x, yin mathscr{E}$ if and only if there exists an element $gamma$ of the center $Z(M({mathscr A}))$ of the multiplier algebra $M({mathscr A})$ of ${mathscr A}$ such that $langle T(x), S(y)rangle = gamma langle x, yrangle$ for all $x, yin mathscr{E}$. In particular, for adjointable operators $S$ we have $T=(S^*)^{-1}$, and any bounded invertible module operator $T$ may appear. Varying the conditions on the mappings $T$ and $S$ we obtain further affirmative results for local operators and for pairs of a bounded and of an unbounded module operator with bounded inverse, among others. Also, unbounded operators with disjoint ranges are considered. The proving techniques give new insights.
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