Relative entropy for pairs of subalgebras and invariants arising from automorphisms

对于算子代数M的子代数对{A，B}，通过修改Connes-Stormer相对熵H(A|B)，定义了h(A|B)，并度量了两个代数之间的距离。例如，在M是大小为n的矩阵代数的情况下，则{h(A，B)；A和B是极大交换子代数M}=[0.logn]且h(A，B)=logn当且仅当A和B相互正交。类似的结果也适用于指数为2的第II_1型因子对。

For a pair {A, B} of subalgebra of an operator algebra M, by modifying the Connes-Stormer relative entropy H(A|B), we defined h(A|B), and measured of distance between two algebras. For examples, in the case where M is the matrix algebra with the size n, then { h(A,B) ; A and B are maximal abelian subalgebras M}=[0.log n] and h(A,B)=log n iff A and B are mutually orthogonal. Similar result holds for pairs of subfactor with index 2 of type II_1 factors.