论文标题
关于预测的有限总和，Dixmier的平均定理类型$ {\ rm ii} _1 $因子
On finite sums of projections and Dixmier's averaging theorem for type ${\rm II}_1$ factors
论文作者
论文摘要
令$ \ mathcal {m} $为类型$ {\ rm ii_1} $ factor，让$τ$为$ \ Mathcal {m} $上的忠实正常奇特状态。在本文中，我们证明了$ x \ in \ mathcal {m} $，$ x = x^*$，然后将身份分解为$ n \ in \ mathbb {n} $相互正交的nonimalotally nol Zero nonno nol Zero Projections $ e_jj \ in \ Mathcal in \ Mathcal {m} $ sum_ $ e_jxe_j =τ（x）e_j $ for asl $ j = 1，\ cdots，n $。等效地，有一个单一操作员$ u \ in \ Mathcal {m} $，带有$ u^n = i $和$ \ frac {1} {n} {n} {n} \ sum_ { \ Mathcal {m} $可以作为$ \ Mathcal {m} $的有限投影总和（如果且仅考虑$τ（a）\ geqτ（r_a）$，其中$ r_a $是$ a $的范围投影。这个结果回答了[9]的第6.7问题。作为第二个应用程序，我们表明，如果$ x \ in \ Mathcal {m} $，$ x = x^*$和$τ（x）= 0 $，则存在一个nilpotent元素$ z \ in \ mathcal {m} $，因此$ x $是$ z $的实际部分。该结果回答了[4]的肯定问题1.1。作为第三个应用程序，我们表明$ x_1，\ cdots，x_n \ in \ mathcal {m} $。然后存在单位运算符$ u_1，\ cdots，u_k \ in \ Mathcal {m} $，以至于$ \ frac {1} {k} {k} {k} \ sum_ {i = 1}^ku_i^{ - 1} x_ju_i = f leq fe f leq fe fe fe fe fe fe fe。该结果是Dixmier的平均定理的更强版本，用于$ {\ rm II} _1 $因子。
Let $\mathcal{M}$ be a type ${\rm II_1}$ factor and let $τ$ be the faithful normal tracial state on $\mathcal{M}$. In this paper, we prove that given an $X \in \mathcal{M}$, $X=X^*$, then there is a decomposition of the identity into $N \in \mathbb{N}$ mutually orthogonal nonzero projections $E_j\in\mathcal{M}$, $I=\sum_{j=1}^NE_j$, such that $E_jXE_j=τ(X) E_j$ for all $j=1,\cdots,N$. Equivalently, there is a unitary operator $U \in \mathcal{M}$ with $U^N=I$ and $\frac{1}{N}\sum_{j=0}^{N-1}{U^*}^jXU^j=τ(X)I.$ As the first application, we prove that a positive operator $A\in \mathcal{M}$ can be written as a finite sum of projections in $\mathcal{M}$ if and only if $τ(A)\geq τ(R_A)$, where $R_A$ is the range projection of $A$. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if $X\in \mathcal{M}$, $X=X^*$ and $τ(X)=0$, then there exists a nilpotent element $Z \in \mathcal{M}$ such that $X$ is the real part of $Z$. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let $X_1,\cdots,X_n\in \mathcal{M}$. Then there exist unitary operators $U_1,\cdots,U_k\in\mathcal{M}$ such that $\frac{1}{k}\sum_{i=1}^kU_i^{-1}X_jU_i=τ(X_j)I,\quad \forall 1\leq j\leq n$. This result is a stronger version of Dixmier's averaging theorem for type ${\rm II}_1$ factors.
