Compactly-aligned discrete product systems, and generalizations of O_\infty

Realising the C*-algebra of a higher-rank graph as an Exel crossed product

We use the boundary-path space of a finitely-aligned k-graph Λto construct a compactly-aligned product system X, and we show that the graph algebra C^*(Λ) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.

积系统的C*-代数的有限维逼近

Finite-dimensional approximations for C*-algebras of product systems

研究积系统生成的C*-代数的精确性与核性，证明其等价于核或系数代数的相应性质，并指出紧对齐假设的必要性。

Colocalizations of noncommutative spectra and bootstrap categories

We construct a compactly generated and closed symmetric monoidal stable $\infty$-category $\mathtt{NSp'}$ and show that $\mathtt{hNSp'}^{op}$ contains the suspension stable homotopy category of separable $C^*$-algebras $\mathtt{ΣHo^{C^*}}$ constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of $\mathtt{NSp'}$, namely, $\mathtt{NSp'}[\mathbb{K}^{-1}]$ and $\mathtt{NSp'}[\mathcal{Z}^{-1}]$, both of which are shown to be compactly generated and closed symmetric monoidal. We prove that Kasparov $KK$-category of separable $C^*$-al…

C*-algebras associated to product systems of Hilbert bimodules

Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems …

拟格序群胚上积系统相关的Cuntz-Nica-Pimsner代数

Cuntz-Nica-Pimsner algebras associated to product systems over quasi-lattice ordered groupoids

本文刻画拟格序群胚上紧对齐积系统的Cuntz-Nica-Pimsner代数，并证明特定条件下Morita等价双模的Fell丛完全截面$C^*$-代数与其同构。

Operator algebras for higher rank analysis and their application to factorial languages

We study strong compactly aligned product systems of $\mathbb{Z}_+^N$ over a C*-algebra $A$. We provide a description of their Cuntz-Nica-Pimsner algebra in terms of tractable relations coming from ideals of $A$. This approach encompasses product systems where the left action is given by compacts, as well as a wide class of higher rank graphs (beyond row-finite). Moreover we analyze higher rank factorial languages and their C*-algebras. Many of the rank one results in the literature find here their higher rank analogues. In particular, we show that the Cuntz-Nica-Pimsner algebra of a higher ra…

Operator algebra quantum groups of universal gauge groups

In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra setting. More precisely, we propose a concise definition of sigma-C*-quantum groups and explain the concept here. At the same time, we put this definition in the mathematical context of countably compactly generated groups as well as C*-compact quantum groups.

An operator that relates to semi-meander polynomials via a two-sided q-Wick formula

We consider the sequence $( Q_n )_{n=1}^{\infty}$ of semi-meander polynomials which are used in the enumeration of semi-meandric systems (a family of diagrams related to the classical stamp-folding problem). We show that for a fixed natural number $d$, the sequence $( Q_n (d) )_{n=1}^{\infty}$ appears as sequence of moments for a compactly supported probability measure $ν_d$ on the real line. More generally, we consider a two-variable generalization $Q_n (t,u)$ of $Q_n(t)$, which is related to a natural concept of "self-intersecting meandric system"; the second variable of $Q_n (t,u)$ keeps tr…

关于乘积系统的Fock协变性与离散作用下的约化Hao-Ng同构问题

On Fock covariance for product systems and the reduced Hao-Ng isomorphism problem by discrete actions

本文刻画了乘积系统的等变Fock协变单射表示，证明其与Nica协变性和Toeplitz表示一致，并据此解决了离散群广义规范作用的约化Hao-Ng同构问题。

Operator algebra quantum homogeneous spaces of universal gauge groups

In this paper, we quantize universal gauge groups such as SU(\infty), as well as their homogeneous spaces, in the sigma-C*-algebra setting. More precisely, we propose concise definitions of sigma-C*-quantum groups and sigma-C*-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C*-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute it for the quantum homogeneous spaces associated to the universal gauge …

Noncommutative CW-spectra as enriched presheaves on matrix algebras

Motivated by the philosophy that $C^*$-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of $C^*$-algebras. We focus on $C^*$-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable $\infty$-category of noncommutative CW-spectra, which we denote by $\mathtt{NSp}$. Let $\mathcal{M}$ be the full spectral subcategory of $\mathtt{NSp}$ spanned by "noncommutative suspension spectra" of matrix algebras. Our main result is that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheave…

右LCM半群上乘积系统的余泛性与受控映射

Co-universality and controlled maps on product systems over right LCM-semigroups

该文研究右LCM半群上紧对齐乘积系统的C*-代数结构，结合受控映射、受控消去法与C*-包络理论，推导了余泛C*-代数刻画、核性与正合性条件及饱和受控映射下的系统性质等结果。