相关论文: Notes on the type classification of von Neumann al…

This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on…

Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is…

We classify C$^*$ near-group categories by using Vaughan Jones theory of subfactors and the Cuntz algebra endomorphisms. Our results show that there is a sharp contrast between two essentially different cases, integral and irrational cases.…

These are introductory lecture notes aimed at beginning graduate students covering fundamental concepts and ideas behind the renormalisation group. Our main goal is to motivate it and then explore its consequences, in the context of quantum…

We study the von Neumann algebra, generated by the regular representations of the infinite-dimensional nilpotent group $B_0^{\mathbb Z}$. In [14] a condition have been found on the measure for the right von Neumann algebra to be the…

This paper includes a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors \cite{CS}, we…

We initiate a study of linear maps on $M_n(\mathbb{C})$ that have the property that they factor through a tracial von Neumann algebra $(\mathcal{A,\tau})$ via operators $Z\in M_n(\mathcal{A})$ whose entries consist of positive elements from…

For each $\lambda\in\left]0,1\right]$ we exhibit an uncountable family of compact quantum groups $\mathbb{G}$ such that the von Neumann algebra $\mathsf{L}^{\!\infty}(\mathbb{G})$ is the injective factor of type $\mathrm{III}_\lambda$ with…

We study quantum information and computation from a novel point of view. Our approach is based on recasting the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of compact closed categories…

This is a first stab at a mathematical framework in which one can study quantum field theories on spacetimes with quite general geometries. We will study these theories via their factorization algebras. The aim is to identify a minimalist…

An inclusion of II$_1$ factors $N \subset M$ with finite Jones index gives rise to a powerful set of invariants that can be approached successfully in a number of different ways. We describe Jones' pictorial description of the standard…

In this article we give a short and informal overview of some aspects of the theory of C*- and von Neumann algebras. We also mention some classical results and applications of these families of operator algebras.

We define type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ as well as C*-semi-finite C*-algebras. It is shown that a von Neumann algebra is a type $\mathfrak{A}$, type $\mathfrak{B}$, type $\mathfrak{C}$ or C*-semi-finite…

Let $n\geq 2$ and $G_n=\mathbb{Z}^n\rtimes SL_n(\mathbb{Z})$. We classify all $G_n$-invariant von Neumann subalgebras in $L(G_n)$. For $n=2$, this gives an alternative proof of the previous result of Jiang-Liu. For $n\geq 3$, this gives the…

We propose a new line of attack to create a finite quantum theory which includes general relativity and (perhaps) the standard model in its low energy limit. The theory would emerge from the categorical approach. A structure is observed on…

We characterize injectivity of von Neumann algebras in terms of factoring bilinear maps as products of linear maps.

As applied to quantum theories, the program of renormalization is successful for `renormalizable models' but fails for `nonrenormalizable models'. After some conceptual discussion and analysis, an enhanced program of renormalization is…

This is a brief review of our recent work attempted at a generalization of the Grassmann algebra to the paragrassmann ones. The main aim is constructing an algebraic basis for representing `fractional' symmetries appearing in $2D$…

Let $M$ be a finite von Neumann algebra. In the first part, we give asymptotic results about $M$-stable sequences of weak*-continuous mappings which are related with operators belonging to $M$. In the second part, we extend, by a shorter…

An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III$_1$, in which entropy cannot be well-defined because such algebras do not admit a trace or density states.…