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Register a new userQuantum smooth uncertainty principles for von Neumann bi-algebras
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Added by
Zhengwei Liu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify numbers of uncertainty principles on quantum symmetries, such as subfactors, and fusion bi-algebras etc, studied in quantum Fourier analysis. We also obtain Widgerson-Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson.
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A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.
In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters $0<p,qleq infty$. Furthermore, we establish R{e}nyi entropic uncertainty principles for subfactor planar algebras.
A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970s by Araki, Connes, Haagerup, Takesaki and others, which are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. The course was aimed at giving a fast track study of those main classics of the theory, from which people gain an enough background knowledge so that they can consult suitable volumes when more details are needed.
We consider the general linear group as an invariant of von Neumann factors. We prove that up to complement, a set consisting of all idempotents generating the same right ideal admits a characterisation in terms of properties of the general linear group of a von Neumann factor. We prove that for two Neumann factors, any bijection of their general linear groups induces a bijection of their idempotents with the following additional property: If two idempotents or their two complements generate the same right ideal, then so does their image. This generalises work on regular rings, such include von Neumann factors of type $I_{n}$, $n < infty$.
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$) be the algebra of all $tau$-measurable (resp. $tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$
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