$\mathbb Z_n$--graded Independence

$\mathbb{Z}_2$分次$C^*$代数的$C^*$独立性

$C^*$-independence for $\mathbb{Z}_2$-graded $C^*$-algebras

本文研究$\mathbb{Z}_2$分次$C^*$代数的$C^*$独立性，建立分次von Neumann代数上多种统计独立性概念的关系，并刻画分次核性质。

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…

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

We provide an operator algebraic proof of a classical theorem of Thoma which characterizes the extremal characters of the infinite symmetric group $\mathbb{S}_\infty$. Our methods are based on noncommutative conditional independence emerging from exchangeability and we reinterpret Thoma's theorem as a noncommutative de Finetti type result. Our approach is, in parts, inspired by Jones' subfactor theory and by Okounkov's spectral proof of Thoma's theorem, and we link them by inferring spectral properties from certain commuting squares.

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.