Compact Kac algebras and commuting squares

交换平方可和矩阵族的正规逼近

Normal approximations of commuting square-summable matrix families

本文证明任意平方可和交换矩阵族都可在由其数值域直径控制的距离内被正规交换族逼近，并由此统一导出 Fraas 的正规性判别结果与接近等距矩阵的经典结论。

Graph planar algebra embeddings and infinite depth subfactors

Subfactors of the hyperfinite II$_1$ factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index $\frac{5+\sqrt{13}}{2}$, i.e. the index of the Haage…

非交换闵可夫斯基积分不等式与融合环的单位ary分类化准则

Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings

本文证明了非交换闵可夫斯基积分不等式，并给出了融合环的单位ary分类化准则。

二维拓扑序中4张量的zipper条件以及源自双单位ary连接的子因子的更高相对交换量

The zipper condition for $4$-tensors in two-dimensional topological order and the higher relative commutants of a subfactor arising from a commuting square

本文研究了二维拓扑序中4张量的zipper条件，并将其与子因子理论中的更高相对交换量联系起来，揭示了双单位ary连接与平坦场之间的对应关系。

Almost commuting matrices with respect to the rank metric

We show that if A_1, A_2, ... , A_n are square matrices, each of them is either unitary or self-adjoint, and they almost commute with respect to the rank metric, then one can find commuting matrices B_1, B_2, ... , B_n that are close to the matrices A_i in the rank metric.

On some fundamental results about higher-rank graphs and their C*-algebras

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated to a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a ho…

von Neumann 子代数之间的距离：自旋模型、交换平方与自由群因子

Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors

本文研究 von Neumann 子代数间的 Mashood-Taylor 与 Kadison-Kastler 距离，证明自旋模型子因子距离公式、交换平方与最大距离的等价性，并在自由群因子中构造最大距离的 masas。

Group-type subfactors and Hadamard matrices

A hyperfinite $II_1$ subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusion $R^H \subset R \rtimes K$, where $H$ and $K$ are finite groups with outer actions on the hyperfinite $II_1$ factor $R$. We find the group of outer automorphisms generated by $H$ and $K$, and use the method of Bisch and Haagerup to determine the principal and dual principal graphs. In some cases a complete classification is obtained by examining the element …

On Square Roots of the Haar State on Compact Quantum Groups

The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphi…

Blends and Alloys

Given two algebras A and B, sometimes assumed to be C*-algebras, we consider the question of putting algebra or C*-algebra structures on the tensor product A\otimes B. In the C*-case, assuming B to be two-dimensonal, we characterize all possible such C*-algebra structures in terms of an action of the cyclic group Z_2. An example related to commuting squares is also discussed.