Zhengwei LiuYau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, 100084, China; Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, 101408, ChinaSebastien PalcouxBeijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, 101408, ChinaJinsong WuInstitute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China
Category TheoryFunctional AnalysisQuantum AlgebraRings and AlgebrasSpectral Theory and Operator Algebramathscidoc:2207.04003
We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.
The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).
To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions,
called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies.
We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines
our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected.
We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification.
As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.
Lu CuiAcademy of Mathematics and Systems Science, Chinese Academy of SciencesLinzhe HuangTsinghua University, YMSCWenming WuSchool of Mathematical Sciences, Chongqing Normal UniversityWei YuanAcademy of Mathematics and Systems Science, Chinese Academy of SciencesHanbin ZhangSchool of Mathematics (Zhuhai), Sun Yat-sen University
Journal of Mathematical Analysis and Applications, 21, 2022.4
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.
Yong LinDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaShuang LiuYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, ChinaHongye SongDepartment of Mathematics, Renmin University of China, Beijing 100872, China; Beijing International Studies University, Beijing 100024, China
We prove the equivalence between some functional inequalities and the ultracontractivity property of the heat semigroup on infinite graphs. These functional inequalities include Sobolev inequalities, Nash inequalities, Faber–Krahn inequalities, and log-Sobolev inequalities. We also show that, under the assumptions of volume growth and CDE(n, 0), which is regarded as the natural notion of curvature on graphs, these four functional inequalities and the ultracontractivity property of the heat semigroup are all true on graphs.