$\mathbb{Z}_2$分次$C^*$代数的$C^*$独立性

$\mathbb Z_n$-分级独立性

$\mathbb Z_n$--graded Independence

本文将Mingo和Nica关于分次独立性的结果从$\mathbb Z_2$-分次（费米）非对易概率空间推广到$\mathbb Z_n$-分次非对易概率空间.我们证明了对于$q$是$n$的本原单位根，由Nica定义的$q$-累积量线性化了齐次$\mathbb Z_n$-分次独立随机变量的加法.

On $C^*$-norms on $\mathbb{Z}_2$-graded tensor products

We systematically investigate $C^*$-norms on the algebraic graded product of $\mathbb{Z}_2$-graded $C^*$-algebras. This requires to single out the notion of a compatible norm, that is a norm with respect to which the product grading is bounded. We then focus on the spatial norm proving that it is minimal among all compatible $C^*$-norms. To this end, we first show that commutative $\mathbb{Z}_2$-graded $C^*$-algebras enjoy a nuclearity property in the category of graded $C^*$-algebras. In addition, we provide a characterization of the extreme even states of a given graded $C^*$-algebra in term…

无限对称群$\mathbb{s}_\infty$的特征标的非交换独立性

Noncommutative Independence from Characters of the Infinite Symmetric Group $\mathbb{s}_\infty$

本文给出了Thoma关于无限对称群$\mathbb{S}_\infty$极值特征标的一个经典定理的算子代数证明。我们的方法是基于非交换的条件独立性出现的exchangecutive和我们重新解释托马的定理作为一个非交换的de Finetti型的结果。我们的方法是，在部分，琼斯的子因子理论和Okounkov的频谱证明托马定理的启发，我们将它们联系起来，从某些交换平方推断频谱特性。

Graded extensions of generalized Haagerup categories

We classify certain $\mathbb{Z}_2 $-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: $\mathbb{Z}_2 $-graded extensions of $\mathbb{Z}_{2n} $ generalized Haagerup categories for all $n \leq 5 $; $\mathbb{Z}_2 \times \mathbb{Z}_2 $-graded extensions of the Asaeda-Haagerup categories; and extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2 $ generalized Haagerup category by its outer automorphism group $A_4 $. The construction uses endomorphis…

$\mathbb{Z}$-graded rings as Cuntz-Pimsner rings

Given a $\mathbb{Z}$-graded ring $A$ and a subring $R\subseteq A$, it is natural to ask whether $A$ can be realised as the Cuntz-Pimsner ring of some $R$-system. In this paper, we derive sufficient conditions on $A$ and $R$ for this to be the case. As an application, we give conditions under which the Steinberg algebra $A_K(\mathcal{G})$ associated to a $\mathbb{Z}$-graded groupoid $\mathcal{G}=\sqcup_{n\in \mathbb{Z}} \mathcal{G}_n$ can be realised as the Cuntz-Pimsner ring of an $A_K(\mathcal{G}_0)$-system.

The graded structure of algebraic Cuntz-Pimsner rings

The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.