Let $\mathscr{M}$ be a $II_1$ factor acting on the Hilbert space $\mathscr{H}$, and $\mathscr{M}_{\textrm{aff}}$ be the Murray-von Neumann algebra of closed densely-defined operators affiliated with $\mathscr{M}$. Let $\tau$ denote the unique faithful normal tracial state on $\mathscr{M}$. By virtue of Nelson's theory of non-commutative integration, $\mathscr{M}_{\textrm{aff}}$ may be identified with the completion of $\mathscr{M}$ in the measure topology. In this article, we show that $M_n(\mathscr{M}_{\textrm{aff}}) \cong M_n(\mathscr{M})_{\textrm{aff}}$ as unital ordered complex topological $*$-algebras with the isomorphism extending the identity mapping of $M_n(\mathscr{M}) \to M_n(\mathscr{M})$. Consequently, the algebraic machinery of rank identities and determinant identities are applicable in this setting. As a step further in the Heisenberg-von Neumann puzzle discussed by Kadison-Liu (SIGMA, 10 (2014), Paper 009), it follows that if there exist operators $P, Q$ in $\mathscr{M}_{\textrm{aff}}$ satisfying the commutation relation $Q \; \hat \cdot \; P \; \hat - \; P \; \hat \cdot \; Q = {i\mkern1mu} I$, then at least one of them does not belong to $L^p(\mathscr{M}, \tau)$ for any $0 < p \le \infty$. Furthermore, the respective point spectrums of $P$ and $Q$ must be empty. Hence the puzzle may be recasted in the following equivalent manner - Are there invertible operators $P, A$ in $\mathscr{M}_{\textrm{aff}}$ such that $P^{-1} \; \hat \cdot \; A \; \hat \cdot \; P = I \; \hat + \; A$? This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in $\mathscr{M}_{\textrm{aff}}$ in an essential way.

中文翻译：

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无界算子代数上的矩阵代数

令 $\mathscr{M}$ 是作用于希尔伯特空间 $\mathscr{H}$ 的 $II_1$ 因子，而 $\mathscr{M}_{\textrm{aff}}$ 是默里-冯诺依曼代数隶属于 $\mathscr{M}$ 的闭合密集定义运算符。让 $\tau$ 表示 $\mathscr{M}$ 上唯一的忠实正常轨迹状态。凭借 Nelson 的非交换积分理论，$\mathscr{M}_{\textrm{aff}}$ 可以与度量拓扑中的 $\mathscr{M}$ 的完成标识。在这篇文章中，我们展示了 $M_n(\mathscr{M}_{\textrm{aff}}) \cong M_n(\mathscr{M})_{\textrm{aff}}$ 作为单序复拓扑 $*具有同构的 $-代数扩展 $M_n(\mathscr{M}) \to M_n(\mathscr{M})$ 的恒等映射。因此，秩恒等式和行列式恒等式的代数机制适用于这种情况。作为 Kadison-Liu (SIGMA, 10 (2014), Paper 009) 讨论的 Heisenberg-von Neumann 谜题的更进一步，如果在 $\mathscr{M}_{\ textrm{aff}}$ 满足交换关系 $Q \; \hat \cdot \; P\; \帽子 - \; P\; \hat \cdot \; Q = {i\mkern1mu} I$，那么对于任何 $0 < p \le \infty$，其中至少有一个不属于 $L^p(\mathscr{M}, \tau)$。此外，$P$ 和$Q$ 各自的点谱必须为空。因此，这个谜题可以用以下等价的方式重铸——在 $\mathscr{M}_{\textrm{aff}}$ 中是否存在可逆运算符 $P, A$ 使得 $P^{-1} \; \hat \cdot \; 一种 \; \hat \cdot \; P = I \; \帽子 + \; 澳元？这表明任何解决该问题的策略都必须以基本方式研究 $\mathscr{M}_{\textrm{aff}}$ 中运算符的共轭不变量。