We give a short and simple proof, utilizing the pre-determinant of P. de la Harpe and G. Skandalis, that the universal covering group of the unitary group of a II$_1$ von Neumann algebra $mathcal{M}$, when equipped with the norm topology, splits algebraically as the direct product of the self-adjoint part of its center and the unitary group $U(mathcal{M})$. Thus, when $mathcal{M}$ is a II$_1$ factor, the universal covering group of $U(mathcal{M})$ is algebraically isomorphic to the direct product $mathbb{R} times U(mathcal{M})$. In particular, the question of P. de la Harpe and D. McDuff of whether the universal cover of $U(mathcal{M})$ is a perfect group is answered in the negative.
我们利用 P. de laHarpe 和 G. Skandalis 的预判定式给出了一个简短的证明：当配备了规范拓扑学时，一个 II$_1$ von Neumann 代数 $mathcal{M}$ 的单元群的普遍覆盖群在代数上分裂为其中心自交部分与单元群 $U(mathcal{M})$ 的直接乘积。因此，当 $mathcal{M}$ 是一个 II$_1$ 因子时，$U(mathcal{M})$ 的普遍覆盖组在代数上与 $mathbb{R} timesU(mathcal{M})$ 的直积同构。特别是，P. de la Harpe 和 D. McDuff 关于 $U(mathcal{M})$ 的普遍盖是否是一个完全群的问题得到了否定的回答。
{"title":"Universal covering groups of unitary groups of von Neumann algebras","authors":"Pawel Sarkowicz","doi":"arxiv-2408.13710","DOIUrl":"https://doi.org/arxiv-2408.13710","url":null,"abstract":"We give a short and simple proof, utilizing the pre-determinant of P. de la\u0000Harpe and G. Skandalis, that the universal covering group of the unitary group\u0000of a II$_1$ von Neumann algebra $mathcal{M}$, when equipped with the norm\u0000topology, splits algebraically as the direct product of the self-adjoint part\u0000of its center and the unitary group $U(mathcal{M})$. Thus, when $mathcal{M}$\u0000is a II$_1$ factor, the universal covering group of $U(mathcal{M})$ is\u0000algebraically isomorphic to the direct product $mathbb{R} times\u0000U(mathcal{M})$. In particular, the question of P. de la Harpe and D. McDuff of\u0000whether the universal cover of $U(mathcal{M})$ is a perfect group is answered\u0000in the negative.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The study of Khintchin inequalities has a long history in abstract harmonic analysis. While there is almost no possibility of non-trivial Khintchine inequality for central Fourier series on compact connected semisimple Lie groups, we demonstrate a strong contrast within the framework of compact quantum groups. Specifically, we establish a Khintchine inequality with operator coefficients for arbitrary central Fourier series in a large class of non-Kac compact quantum groups. The main examples include the Drinfeld-Jimbo $q$-deformations $G_q$, the free orthogonal quantum groups $O_F^+$, and the quantum automorphism group $G_{aut}(B,psi)$ with a $delta$-form $psi$.
{"title":"A Khintchine inequality for central Fourier series on non-Kac compact quantum groups","authors":"Sang-Gyun Youn","doi":"arxiv-2408.13519","DOIUrl":"https://doi.org/arxiv-2408.13519","url":null,"abstract":"The study of Khintchin inequalities has a long history in abstract harmonic\u0000analysis. While there is almost no possibility of non-trivial Khintchine\u0000inequality for central Fourier series on compact connected semisimple Lie\u0000groups, we demonstrate a strong contrast within the framework of compact\u0000quantum groups. Specifically, we establish a Khintchine inequality with\u0000operator coefficients for arbitrary central Fourier series in a large class of\u0000non-Kac compact quantum groups. The main examples include the Drinfeld-Jimbo\u0000$q$-deformations $G_q$, the free orthogonal quantum groups $O_F^+$, and the\u0000quantum automorphism group $G_{aut}(B,psi)$ with a $delta$-form $psi$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper establishes a necessary and sufficient condition for the coincidence of non-commutative $log$-algebras constructed from different exact normal semifinite traces. Consequently, we provide a criterion for the isomorphism of $log$-algebras built on non-commutative von Neumann algebras with different exact normal semifinite traces. Additionally, we demonstrate a connection between the isomorphism of non-commutative $log$-algebras and the isomorphism of the corresponding $log$-algebras constructed on the center of these von Neumann algebras. Furthermore, we present a necessary and sufficient condition for the isomorphism of $log$-algebras derived from different von Neumann algebras of type $I_n$.
本文为由不同的精确正态半有限迹所构造的非交换$log$-gebras的共存建立了一个必要条件和充分条件。因此，我们为建立在具有不同精确正态半有限迹的非交换冯-诺依曼代数上的＄log＄-代数的同构提供了一个标准。此外，我们证明了非交换 $log$-gebras 的同构性与在这些 von Neumann 对象的中心上构造的相应 $log$-gebras 的同构性之间的联系。此外，我们还提出了从 $I_n$ 类型的不同 von Neumann 对象衍生的 $log$ 对象的同构性的必要和充分条件。
{"title":"The inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $log$-algebras","authors":"Rustam Abdullaev, Azizkhon Azizov","doi":"arxiv-2408.13527","DOIUrl":"https://doi.org/arxiv-2408.13527","url":null,"abstract":"This paper establishes a necessary and sufficient condition for the\u0000coincidence of non-commutative $log$-algebras constructed from different exact\u0000normal semifinite traces. Consequently, we provide a criterion for the\u0000isomorphism of $log$-algebras built on non-commutative von Neumann algebras\u0000with different exact normal semifinite traces. Additionally, we demonstrate a\u0000connection between the isomorphism of non-commutative $log$-algebras and the\u0000isomorphism of the corresponding $log$-algebras constructed on the center of\u0000these von Neumann algebras. Furthermore, we present a necessary and sufficient\u0000condition for the isomorphism of $log$-algebras derived from different von\u0000Neumann algebras of type $I_n$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A quasi-representation of a group is a map from the group into a matrix algebra (or similar object) that approximately satisfies the relations needed to be a representation. Work of many people starting with Kazhdan and Voiculescu, and recently advanced by Dadarlat, Eilers-Shulman-So{}rensen and others, has shown that there are topological obstructions to approximating unitary quasi-representations of groups by honest representations, where `approximation' is understood to be with respect to the operator norm. The purpose of this paper is to explore whether approximation is possible if the known obstructions vanish, partially generalizing work of Gong-Lin and Eilers-Loring-Pedersen for the free abelian group of rank two, and the Klein bottle group. We show that this is possible, at least in a weak sense, for some `low-dimensional' groups including fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three manifolds. The techniques used in the paper are $K$-theoretic: they have their origin in Baum-Connes-Kasparov type assembly maps, and in the Elliott program to classify $C^*$-algebras; Kasparov's bivariant KK-theory is a crucial tool. The key new technical ingredients are: a stable uniqueness theorem in the sense of Dadarlat-Eilers and Lin that works for non-exact $C^*$-algebras; and an analysis of maps on $K$-theory with finite coefficients in terms of the relative eta invariants of Atiyah-Patodi-Singer. Despite the proofs going through $K$-theoretic machinery, the main theorems can be stated in elementary terms that do not need any $K$-theory.
{"title":"Conditional representation stability, classification of $*$-homomorphisms, and eta invariants","authors":"Rufus Willett","doi":"arxiv-2408.13350","DOIUrl":"https://doi.org/arxiv-2408.13350","url":null,"abstract":"A quasi-representation of a group is a map from the group into a matrix\u0000algebra (or similar object) that approximately satisfies the relations needed\u0000to be a representation. Work of many people starting with Kazhdan and\u0000Voiculescu, and recently advanced by Dadarlat, Eilers-Shulman-So{}rensen and\u0000others, has shown that there are topological obstructions to approximating\u0000unitary quasi-representations of groups by honest representations, where\u0000`approximation' is understood to be with respect to the operator norm. The purpose of this paper is to explore whether approximation is possible if\u0000the known obstructions vanish, partially generalizing work of Gong-Lin and\u0000Eilers-Loring-Pedersen for the free abelian group of rank two, and the Klein\u0000bottle group. We show that this is possible, at least in a weak sense, for some\u0000`low-dimensional' groups including fundamental groups of closed surfaces,\u0000certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental\u0000groups of three manifolds. The techniques used in the paper are $K$-theoretic: they have their origin in\u0000Baum-Connes-Kasparov type assembly maps, and in the Elliott program to classify\u0000$C^*$-algebras; Kasparov's bivariant KK-theory is a crucial tool. The key new\u0000technical ingredients are: a stable uniqueness theorem in the sense of\u0000Dadarlat-Eilers and Lin that works for non-exact $C^*$-algebras; and an\u0000analysis of maps on $K$-theory with finite coefficients in terms of the\u0000relative eta invariants of Atiyah-Patodi-Singer. Despite the proofs going\u0000through $K$-theoretic machinery, the main theorems can be stated in elementary\u0000terms that do not need any $K$-theory.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We take a first step towards developing a new language to describe causal structure, event horizons, and quantum extremal surfaces (QES) for the bulk description of holographic systems beyond the standard Einstein gravity regime. By considering the structure of boundary operator algebras, we introduce a stringy ``causal depth parameter'', which quantifies the depth of the emergent radial direction in the bulk, and a certain notion of ergodicity on the boundary. We define stringy event horizons in terms of the half-sided inclusion property, which is related to a stronger notion of boundary ergodic or quantum chaotic behavior. Using our definition, we argue that above the Hawking--Page temperature, there is an emergent sharp horizon structure in the large $N$ limit of $mathcal{N}=4$ Super-Yang--Mills at finite nonzero 't Hooft coupling. In contrast, some previously considered toy models of black hole information loss do not have a stringy horizon. Our methods can also be used to probe violations of the equivalence principle for the bulk gravitational system, and to explore aspects of stringy nonlocality.
{"title":"Toward stringy horizons","authors":"Elliott Gesteau, Hong Liu","doi":"arxiv-2408.12642","DOIUrl":"https://doi.org/arxiv-2408.12642","url":null,"abstract":"We take a first step towards developing a new language to describe causal\u0000structure, event horizons, and quantum extremal surfaces (QES) for the bulk\u0000description of holographic systems beyond the standard Einstein gravity regime.\u0000By considering the structure of boundary operator algebras, we introduce a\u0000stringy ``causal depth parameter'', which quantifies the depth of the emergent\u0000radial direction in the bulk, and a certain notion of ergodicity on the\u0000boundary. We define stringy event horizons in terms of the half-sided inclusion\u0000property, which is related to a stronger notion of boundary ergodic or quantum\u0000chaotic behavior. Using our definition, we argue that above the Hawking--Page\u0000temperature, there is an emergent sharp horizon structure in the large $N$\u0000limit of $mathcal{N}=4$ Super-Yang--Mills at finite nonzero 't Hooft coupling.\u0000In contrast, some previously considered toy models of black hole information\u0000loss do not have a stringy horizon. Our methods can also be used to probe\u0000violations of the equivalence principle for the bulk gravitational system, and\u0000to explore aspects of stringy nonlocality.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the Banach *-algebra $ell^1(G,A,alpha)$, arising from a C*-dynamical system $(A,G,alpha)$, is an hermitian Banach algebra if the discrete group $G$ is finite or abelian (or more generally, a finite extension of a nilpotent group). As a corollary, we obtain that $ell^1(mathbb{Z},C(X),alpha)$ is hermitian, for every topological dynamical system $Sigma = (X, sigma)$, where $sigma: Xto X$ is a homeomorphism of a compact Hausdorff space $X$ and the action is $alpha_n(f)=fcirc sigma^{-n}$ with $ninmathbb{Z}$.
{"title":"Hermitian crossed product Banach algebras","authors":"Rachid El Harti, Paulo R. Pinto","doi":"arxiv-2408.11466","DOIUrl":"https://doi.org/arxiv-2408.11466","url":null,"abstract":"We show that the Banach *-algebra $ell^1(G,A,alpha)$, arising from a\u0000C*-dynamical system $(A,G,alpha)$, is an hermitian Banach algebra if the\u0000discrete group $G$ is finite or abelian (or more generally, a finite extension\u0000of a nilpotent group). As a corollary, we obtain that $ell^1(mathbb{Z},C(X),alpha)$ is hermitian,\u0000for every topological dynamical system $Sigma = (X, sigma)$, where $sigma:\u0000Xto X$ is a homeomorphism of a compact Hausdorff space $X$ and the action is\u0000$alpha_n(f)=fcirc sigma^{-n}$ with $ninmathbb{Z}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a non-invertible dynamical system with a transfer operator, we show there is a minimal cover with a transfer operator that preserves continuous functions. We also introduce an essential cover with even stronger continuity properties. For one-sided sofic subshifts, this generalizes the Krieger and Fischer covers, respectively. Our construction is functorial in the sense that certain equivariant maps between dynamical systems lift to equivariant maps between their covers, and these maps also satisfy better regularity properties. As applications, we identify finiteness conditions which ensure that the thermodynamic formalism is valid for the covers. This establishes the thermodynamic formalism for a large class of non-invertible dynamical systems, e.g. certain piecewise invertible maps. When applied to semi-'etale groupoids, our minimal covers produce 'etale groupoids which are models for $C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers are unified under the common framework of topological graphs.
{"title":"Minimal covers with continuity-preserving transfer operators for topological dynamical systems","authors":"Kevin Aguyar Brix, Jeremy B. Hume, Xin Li","doi":"arxiv-2408.11917","DOIUrl":"https://doi.org/arxiv-2408.11917","url":null,"abstract":"Given a non-invertible dynamical system with a transfer operator, we show\u0000there is a minimal cover with a transfer operator that preserves continuous\u0000functions. We also introduce an essential cover with even stronger continuity\u0000properties. For one-sided sofic subshifts, this generalizes the Krieger and\u0000Fischer covers, respectively. Our construction is functorial in the sense that\u0000certain equivariant maps between dynamical systems lift to equivariant maps\u0000between their covers, and these maps also satisfy better regularity properties.\u0000As applications, we identify finiteness conditions which ensure that the\u0000thermodynamic formalism is valid for the covers. This establishes the\u0000thermodynamic formalism for a large class of non-invertible dynamical systems,\u0000e.g. certain piecewise invertible maps. When applied to semi-'etale groupoids,\u0000our minimal covers produce 'etale groupoids which are models for\u0000$C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers\u0000are unified under the common framework of topological graphs.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David Gao, Srivatsav Kunnawalkam Elayavalli, Mahan Mj
In this note we study a family of graphs of groups over arbitrary base graphs where all vertex groups are isomorphic to a fixed countable sofic group $G$, and all edge groups $H
{"title":"On soficity for certain fundamental groups of graphs of groups","authors":"David Gao, Srivatsav Kunnawalkam Elayavalli, Mahan Mj","doi":"arxiv-2408.11724","DOIUrl":"https://doi.org/arxiv-2408.11724","url":null,"abstract":"In this note we study a family of graphs of groups over arbitrary base graphs\u0000where all vertex groups are isomorphic to a fixed countable sofic group $G$,\u0000and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are\u0000identical everywhere. We prove soficity for this family of groups under a\u0000flexible technical hypothesis for $H$ called $sigma$-co-sofic. This proves\u0000soficity for group doubles $*_H G$, where $H<G$ is an arbitrary separable\u0000subgroup and $G$ is countable and sofic. This includes arbitrary finite index\u0000group doubles of sofic groups among various other examples.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$ to define a height of points of $V(k)$. The corresponding counting function is calculated and we show that it coincides with the known formulas for $n=1$. As an application, it is proved that the set $V(k)$ is finite, whenever the sum of the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the $n$-dimensional Minkowski question-mark function studied by Panti and others.
{"title":"Remark on height functions","authors":"Igor V. Nikolaev","doi":"arxiv-2408.12020","DOIUrl":"https://doi.org/arxiv-2408.12020","url":null,"abstract":"Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety\u0000over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$\u0000to define a height of points of $V(k)$. The corresponding counting function is\u0000calculated and we show that it coincides with the known formulas for $n=1$. As\u0000an application, it is proved that the set $V(k)$ is finite, whenever the sum of\u0000the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the\u0000$n$-dimensional Minkowski question-mark function studied by Panti and others.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define the Cartesian, Categorical, and Lexicographic, and Strong products of quantum graphs. We provide bounds on the quantum chromatic number of these products in terms of the quantum chromatic number of the factors. To adequately describe bounds on the lexicographic product of quantum graphs, we provide a notion of a quantum $b$-fold chromatic number for quantum graphs.
{"title":"Quantum chromatic numbers of products of quantum graphs","authors":"Rolando de Santiago, A. Meenakshi McNamara","doi":"arxiv-2408.11911","DOIUrl":"https://doi.org/arxiv-2408.11911","url":null,"abstract":"We define the Cartesian, Categorical, and Lexicographic, and Strong products\u0000of quantum graphs. We provide bounds on the quantum chromatic number of these\u0000products in terms of the quantum chromatic number of the factors. To adequately\u0000describe bounds on the lexicographic product of quantum graphs, we provide a\u0000notion of a quantum $b$-fold chromatic number for quantum graphs.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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