论文标题
可衡量的操作员的代数是直接有限的
The algebra of thin measurable operators is directly finite
论文作者
论文摘要
让$ \ Mathcal {M} $为Hilbert Space $ \ Mathcal {H} $的半五届von Neumann代数，配备了忠实的普通半限值$τ$，$ S（\ Mathcal {M}，τ），是$ {}^*$ - $ - al $ - 令$ s_0（\ nathcal {m}，τ）$为$ {}^*$ - 所有$τ$ -compact操作员的algebra和$ t（\ nathcal {m}，τ），τ）= s_0（\ s_0） $ x = a+λi$，在s_0（\ mathcal {m}，τ），τ）$和$λ\ in \ mathbb {c} $ in s_0（\ mathcal {m}，τ）中。我们证明，在$ t（\ Mathcal {m}，τ）$中可将$ t（\ Mathcal {m}，τ）$的每个操作员实际上在$ t（\ Mathcal {M}，τ）中可逆。这是对$ \ Mathcal {b}（\ Mathcal {H}）$的薄操作员的薄操作员的子代数的Sterling Berberian Theorem（1982）的概括。对于单数值函数$μ（t; q）$ q = q = q = q^2 \在s（\ mathcal {m}，τ），τ）$中，我们有$μ（t; q）\ in \ {0 \} \ bigcup [1， +\ bigcup [1， +\ f \ \ f \ \ f in $ t> 0 $。它对Daniyar Mushtari在2010年提出的问题给出了积极的答案。
Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\mathcal{H}$ equipped with a faithful normal semifinite trace $τ$, $S(\mathcal{M},τ)$ be the ${}^*$-algebra of all $τ$-measurable operators. Let $S_0(\mathcal{M},τ)$ be the ${}^*$-algebra of all $τ$-compact operators and $T(\mathcal{M},τ)=S_0(\mathcal{M},τ)+\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+λI$ with $A\in S_0(\mathcal{M},τ)$ and $λ\in \mathbb{C}$. We prove that every operator of $T(\mathcal{M},τ)$ that is left-invertible in $T(\mathcal{M},τ)$ is in fact invertible in $T(\mathcal{M},τ)$. It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\mathcal{B} (\mathcal{H})$. For the singular value function $μ(t; Q)$ of $Q=Q^2\in S(\mathcal{M},τ)$ we have $μ(t; Q)\in \{0\}\bigcup [1, +\infty)$ for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.
