Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $τ$ and $E$ be a strongly symmetric Banach function space on $[0,τ(1))$. We show that an operator $x$ in the unit sphere of $E\left(\mathcal{M},τ\right)$ is $k$-extreme, $k\in\mathbb N$, whenever its singular value function $μ(x)$ is $k$-extreme and one of the following conditions hold (i) $μ(\infty,x)=\lim_{t\to\infty}μ(t,x)=0$ or (ii) $n(x)\mathcal{M} n(x^*)=0$ and $|x|\geq μ(\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. The converse is true whenever $\mathcal{M}$ is non-atomic. The global $k$-rotundity property follows, that is if $\mathcal{M}$ is non-atomic then $E$ is $k$-rotund if and only if $E\left(\mathcal{M},τ\right)$ is $k$-rotund. As a consequence of the noncommutive results we obtain that $f$ is a $k$-extreme point of the unit ball of the strongly symmetric function space $E$ if and only if its decreasing rearrangement $μ(f)$ is $k$-extreme and $|f|\geq μ(\infty,f)$. We conclude with the corollary on orbits $Ω(g)$ and $Ω'(g)$. We get that $f$ is a $k$-extreme point of the orbit $Ω(g)$, $g\in L_1+L_{\infty}$, or $Ω'(g)$, $g\in L_1[0,α)$, $α<\infty$, if and only if $μ(f)=μ(g)$ and $|f|\geq μ(\infty,f)$. From this we obtain a characterization of $k$-extreme points in Marcinkiewicz spaces.
Fuglede theorem for symmetric spaces of $τ$-measurable operators
School of Mathematics and Statistics, UNSW（新南威尔士大学数学与统计学院）；Inha University in Tashkent（塔什干英亚大学）；V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan（乌兹别克斯坦科学院V.I.罗曼诺夫斯基数学研究所）
Ergodic theorems in fully symmetric spaces of $τ-$measurable operators
在[11]中，利用非交换插值技巧，建立了非交换Lp-空间（1 < p <∞）中的一个极大遍历定理，并导出了相应的极大遍历不等式和个体遍历定理.本文直接从[25]中导出了非交换Lp-空间中的极大遍历不等式，并应用它们证明了相应的个体和Besicovitch加权遍历定理.然后将这些结果推广到具有Fatou性质和非平凡Boyd指标的非对易全对称巴拿赫空间，特别是推广到非对易Lorentz空间Lpq.研究了非交换全对称巴拿赫空间中遍历平均的范数收敛性。
Form Inequalities for Symmetric Contraction Semigroups
考虑-对于在某个测度空间$\mathrm{X}$，$1\le p < \infty$，$q$上的对称压缩半群的生成元\({-}A\)，对偶指数和给定的可测函数$F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ -陈述：$$ \mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0 $$ { \em for all $\mathbb{C}^d$ -valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in \mathrm{L}^q(\mathrm{X})$ for all $j$ .}证明了这一命题在一般情况下成立，如果$\mathrm{X}$是两点Bernoulli $(\frac{1}{2}, \frac{1}{2})$ -空间，$A$是一种特殊形式.由此得到了这类半群的$\mathrm{L}^{p}$ -解析性的最优角的一个新的证明，它与著名的次马尔可夫情形基本相同.主要定理的证明是结合了著名的约化技巧和$…
