有限维$C^*$代数的Cartan包含的完全刻画
A Complete Characterization of Cartan Inclusions of Finite Dimensional $C^*$-algebras
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中文总结 AI 辅助
该研究根据包含矩阵给出有限维$C^*$代数Cartan包含的完全刻画，证明其等价于无重数矩阵条件或存在唯一条件期望。
中文摘要 AI 辅助
我们根据包含矩阵对有限维$C^*$代数的Cartan包含给出完全刻画。更准确地说，对于包含矩阵为$\boldsymbol{\textit{\u039b}}=(\boldsymbol{\textit{\u039b}}_{ij})$的单位包含$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$，其中Cartan指$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$是Exel意义下的广义Cartan子代数，我们证明该包含为Cartan当且仅当对每个$j$，$\boldsymbol{\textit{\u03a3}}_i \boldsymbol{\textit{\u039b}}_{ij}\boldsymbol{\u2264}1$。我们称满足该条件的矩阵为无重数矩阵。因此，我们的刻画为判定有限维包含是否为Cartan提供了纯组合判据。我们进一步证明，每个Cartan包含都存在从$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$到$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$的唯一条件期望。反之，我们证明对于有限维$C^*$代数的单位包含，条件期望的唯一性足以保证该包含为Cartan。因此，有限维$C^*$代数的单位包含$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}\boldsymbol{\u2286}\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$为Cartan当且仅当存在从$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$到$\boldsymbol{\textit{\u001d}}\boldsymbol{\textit{\u001c}}\boldsymbol{\textit{\u001a}}\boldsymbol{\textit{\u001b}}\boldsymbol{\textit{\u001f}}$的唯一条件期望。
英文摘要
We give a complete characterization of Cartan inclusions of finite dimensional $C^*$-algebras in terms of their inclusion matrices. More precisely, for a unital inclusion $\mathcal{B}\subseteq\mathcal{A}$ with inclusion matrix $Λ=(Λ_{ij})$, where Cartan means that $\mathcal{B}$ is a \emph{generalised Cartan subalgebra} of $\mathcal{A}$ in the sense of Exel, we prove that the inclusion is Cartan if and only if \[ \sum_i Λ_{ij}\leq 1 \] for every $j$. We call matrices satisfying this condition \emph{multiplicity free}. Thus, our characterization provides a purely combinatorial criterion for determining when a finite dimensional inclusion is Cartan. We further prove that every Cartan inclusion admits a unique conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$. Conversely, we show that, for unital inclusions of finite dimensional $C^*$-algebras, the uniqueness of the conditional expectation is sufficient for the inclusion to be Cartan. Consequently, a unital inclusion $\mathcal{B}\subseteq\mathcal{A}$ of finite dimensional $C^*$-algebras is Cartan if and only if there exists a unique conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$.
