标题： 有限维包含的酉正交基 显示英文标题

标题： Unitary orthonormal bases of finite dimensional inclusions

摘要： 我们研究Pimsner和Popa意义上的包含关系$(\mathcal{B}\subseteq \mathcal{A}, E),$的酉正交基，其中$\mathcal{A}, \mathcal{B}$是有限维冯诺依曼代数，$E$是从$\mathcal{A}$到$\mathcal{B}$的条件期望映射。 结果显示，此类基的存在要求相关的包含矩阵满足一个谱条件，迫使维度向量成为Perron-Frobenius特征向量，并且条件期望映射保持马尔可夫迹。 在这些条件下，如果其中一个代数是交换的或单的，则可以显式构造酉正交基。 它们推广了复数Hadamard矩阵、Weyl酉基，以及Crann等人最近的工作，后者分别对应于$\mathcal{A}$为交换代数、单代数和一般多矩阵代数的情况，其中$\mathcal{B}$是复数代数。 首次将$\mathcal{B}$予以推广。 作为这些结果的应用，证明了如果$(\mathcal{B}\subseteq \mathcal{A}, E),$允许一个酉正交基，则 Connes-St{\o }rmer 相对熵$H(\mathcal{A}_1|\mathcal{A})$等于包含矩阵范数平方的对数，其中$\mathcal{A}_1$表示包含的基本构造。 作为进一步的应用，我们证明了对于具有阿贝尔相对交换子的深度2子因子的一大类，存在酉正交基。 显示英文摘要

摘要： We study unitary orthonormal bases in the sense of Pimsner and Popa for inclusions $(\mathcal{B}\subseteq \mathcal{A}, E),$ where $\mathcal{A}, \mathcal{B}$ are finite dimensional von Neumann algebras and $E$ is a conditional expectation map from $\mathcal{A}$ onto $\mathcal{B}$. It is shown that existence of such bases requires that the associated inclusion matrix satisfies a spectral condition forcing dimension vectors to be Perron-Frobenius eigenvectors and the conditional expectation map preserves the Markov trace. Subject to these conditions, explicit unitary orthonormal bases are constructed if either one of the algebras is abelian or simple. They generalize complex Hadamard matrices, Weyl unitary bases, and a recent work of Crann et al which correspond to the special cases of $\mathcal{A}$ being abelian, simple, and general multi-matrix algebras respectively with $\mathcal{B}$ being the algebra of complex numbers. For the first time $\mathcal{B}$ is more general. As an application of these results it is shown that if $(\mathcal{B}\subseteq \mathcal{A}, E),$ admits a unitary orthonormal basis then the Connes-St{\o}rmer relative entropy $H(\mathcal{A}_1|\mathcal{A})$ equals the logarithm of the square of the norm of the inclusion matrix, where $\mathcal{A}_1$ denotes the Jones basic construction of the inclusion. As a further application, we prove the existence of unitary orthonormal bases for a large class of depth 2 subfactors with abelian relative commutant.