In this article we introduce several new examples of Wiener pairs $mathcal{A} subseteq mathcal{B}$, where $mathcal{B} = mathcal{B}(ell^2(X;mathcal{H}))$ is the Banach algebra of bounded operators acting on the Hilbert space-valued Bochner sequence space $ell^2(X;mathcal{H})$ and $mathcal{A} = mathcal{A}(X)$ is a Banach algebra consisting of operator-valued matrices indexed by some relatively separated set $X subset mathbb{R}^d$. In particular, we introduce $mathcal{B}(mathcal{H})$-valued versions of the Jaffard algebra, of certain weighted Schur-type algebras, of Banach algebras which are defined by more general off-diagonal decay conditions than polynomial decay, of weighted versions of the Baskakov-Gohberg-Sj"ostrand algebra, and of anisotropic variations of all of these matrix algebras, and show that they are inverse-closed in $mathcal{B}(ell^2(X;mathcal{H}))$. In addition, we obtain that each of these Banach algebras is symmetric.
{"title":"Wiener pairs of Banach algebras of operator-valued matrices","authors":"Lukas Köhldorfer, Peter Balazs","doi":"arxiv-2407.16416","DOIUrl":"https://doi.org/arxiv-2407.16416","url":null,"abstract":"In this article we introduce several new examples of Wiener pairs\u0000$mathcal{A} subseteq mathcal{B}$, where $mathcal{B} =\u0000mathcal{B}(ell^2(X;mathcal{H}))$ is the Banach algebra of bounded operators\u0000acting on the Hilbert space-valued Bochner sequence space\u0000$ell^2(X;mathcal{H})$ and $mathcal{A} = mathcal{A}(X)$ is a Banach algebra\u0000consisting of operator-valued matrices indexed by some relatively separated set\u0000$X subset mathbb{R}^d$. In particular, we introduce\u0000$mathcal{B}(mathcal{H})$-valued versions of the Jaffard algebra, of certain\u0000weighted Schur-type algebras, of Banach algebras which are defined by more\u0000general off-diagonal decay conditions than polynomial decay, of weighted\u0000versions of the Baskakov-Gohberg-Sj\"ostrand algebra, and of anisotropic\u0000variations of all of these matrix algebras, and show that they are\u0000inverse-closed in $mathcal{B}(ell^2(X;mathcal{H}))$. In addition, we obtain\u0000that each of these Banach algebras is symmetric.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784057","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 study Deaconu-Renault groupoids corresponding to surjective local homeomorphisms on locally compact, Hausdorff, second countable, totally disconnected spaces, and we characterise when the C*-algebras of these groupoids are AF embeddable. Our main result generalises theorems in the literature for graphs and for crossed products of commutative C*-algebras by the integers. We give a condition on the surjective local homeomorphism that characterises the AF embeddability of the C*-algebra of the associated Deaconu-Renault groupoid. In order to prove our main result, we analyse homology groups for AF groupoids, and we prove a theorem that gives an explicit formula for the isomorphism of these groups and the corresponding K-theory. This isomorphism generalises Farsi, Kumjian, Pask, Sims (M"unster J. Math, 2019) and Matui (Proc. Lond. Math. Soc, 2012), since we give an explicit formula for the isomorphism and we show that it preserves positive elements.
我们研究了与局部紧凑、豪斯多夫、第二可数、完全不相连空间上的投射局部同构相对应的 Deaconu-Renault 群组，并描述了当这些群组的 C* 算法是 AF 可嵌入时的特征。我们的主要结果概括了文献中关于图和整数交换 C* 对象的交叉积的定理。我们给出了一个条件，即描述关联的德卡努-雷诺群的 C* 代数的 AF 可嵌入性的射出局部同构。为了证明我们的主要结果，我们分析了AF群的同构群，并证明了一个定理，给出了这些群和相应K理论的同构的明确公式。这个同构概括了Farsi, Kumjian, Pask, Sims (M"unster J. Math, 2019) 和Matui (Proc. Lond. Math. Soc, 2012)，因为我们给出了同构的明确公式，并证明它保留了正元素。
{"title":"AF Embeddability of the C*-Algebra of a Deaconu-Renault Groupoid","authors":"Rafael Pereira Lima","doi":"arxiv-2407.16510","DOIUrl":"https://doi.org/arxiv-2407.16510","url":null,"abstract":"We study Deaconu-Renault groupoids corresponding to surjective local\u0000homeomorphisms on locally compact, Hausdorff, second countable, totally\u0000disconnected spaces, and we characterise when the C*-algebras of these\u0000groupoids are AF embeddable. Our main result generalises theorems in the\u0000literature for graphs and for crossed products of commutative C*-algebras by\u0000the integers. We give a condition on the surjective local homeomorphism that\u0000characterises the AF embeddability of the C*-algebra of the associated\u0000Deaconu-Renault groupoid. In order to prove our main result, we analyse\u0000homology groups for AF groupoids, and we prove a theorem that gives an explicit\u0000formula for the isomorphism of these groups and the corresponding K-theory.\u0000This isomorphism generalises Farsi, Kumjian, Pask, Sims (M\"unster J. Math,\u00002019) and Matui (Proc. Lond. Math. Soc, 2012), since we give an explicit\u0000formula for the isomorphism and we show that it preserves positive elements.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786103","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}
For an action of a discrete group $Gamma$ on a set $X$, we show that the Schreier graph on $X$ is property A if and only if the permutation representation on $ell_2X$ generates an exact $mathrm{C}^*$-algebra. This is well known in the case of the left regular action on $X=Gamma$. This also generalizes Sako's theorem, which states that exactness of the uniform Roe algebra $mathrm{C}^*_{mathrm{u}}(X)$ characterizes property A of $X$ when $X$ is uniformly locally finite.
{"title":"$mathrm{C}^*$-exactness and property A for group actions","authors":"Hiroto Nishikawa","doi":"arxiv-2407.16130","DOIUrl":"https://doi.org/arxiv-2407.16130","url":null,"abstract":"For an action of a discrete group $Gamma$ on a set $X$, we show that the\u0000Schreier graph on $X$ is property A if and only if the permutation\u0000representation on $ell_2X$ generates an exact $mathrm{C}^*$-algebra. This is\u0000well known in the case of the left regular action on $X=Gamma$. This also\u0000generalizes Sako's theorem, which states that exactness of the uniform Roe\u0000algebra $mathrm{C}^*_{mathrm{u}}(X)$ characterizes property A of $X$ when $X$\u0000is uniformly locally finite.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784131","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 prove that a C$^*$-algebra $A$ has uniform property $Gamma$ if the set of extremal tracial states, $partial_e T(A)$, is a non-empty compact space of finite covering dimension and for each $tau in partial_e T(A)$, the von Neumann algebra $pi_tau(A)''$ arising from the GNS representation has property $Gamma$.
{"title":"Uniform property $Γ$ and finite dimensional tracial boundaries","authors":"Samuel Evington, Christopher Schafhauser","doi":"arxiv-2407.16612","DOIUrl":"https://doi.org/arxiv-2407.16612","url":null,"abstract":"We prove that a C$^*$-algebra $A$ has uniform property $Gamma$ if the set of\u0000extremal tracial states, $partial_e T(A)$, is a non-empty compact space of\u0000finite covering dimension and for each $tau in partial_e T(A)$, the von\u0000Neumann algebra $pi_tau(A)''$ arising from the GNS representation has\u0000property $Gamma$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786343","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 prove that the space of traces $text{T}(A)$ of the unital full free product $A=A_1*A_2$ of two unital, separable $C^*$-algebras $A_1$ and $A_2$ is typically a Poulsen simplex, i.e., a simplex whose extreme points are dense. We deduce that $text{T}(A)$ is a Poulsen simplex whenever $A_1$ and $A_2$ have no $1$-dimensional representations, e.g., if $A_1$ and $A_2$ are finite dimensional with no $1$-dimensional direct summands. Additionally, we characterize when the space of traces of a free product of two countable groups is a Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_1,M_2)$ of a tracial von Neumann algebra $M$ which gives necessary conditions ensuring that $M_1$ and a small unitary perturbation of $M_2$ generate a II$_1$ factor.
{"title":"Trace spaces of full free product $C^*$-algebras","authors":"Adrian Ioana, Pieter Spaas, Itamar Vigdorovich","doi":"arxiv-2407.15985","DOIUrl":"https://doi.org/arxiv-2407.15985","url":null,"abstract":"We prove that the space of traces $text{T}(A)$ of the unital full free\u0000product $A=A_1*A_2$ of two unital, separable $C^*$-algebras $A_1$ and $A_2$ is\u0000typically a Poulsen simplex, i.e., a simplex whose extreme points are dense. We\u0000deduce that $text{T}(A)$ is a Poulsen simplex whenever $A_1$ and $A_2$ have no\u0000$1$-dimensional representations, e.g., if $A_1$ and $A_2$ are finite\u0000dimensional with no $1$-dimensional direct summands. Additionally, we\u0000characterize when the space of traces of a free product of two countable groups\u0000is a Poulsen simplex. Our main technical contribution is a new perturbation\u0000result for pairs of von Neumann subalgebras $(M_1,M_2)$ of a tracial von\u0000Neumann algebra $M$ which gives necessary conditions ensuring that $M_1$ and a\u0000small unitary perturbation of $M_2$ generate a II$_1$ factor.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784059","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}
Using Kirchberg-Phillips' classification of purely infinite C*-algebras by K-theory, we prove that the isomorphism types of crossed product C*-algebras associated to certain hyperbolic 3-manifold groups acting on their Gromov boundary only depend on the manifold's homology. As a result, we obtain infinitely many pairwise non-isomorphic hyperbolic groups all of whose associated crossed products are isomorphic. These isomomorphisms are not of dynamical nature in the sense that they are not induced by isomorphisms of the underlying groupoids.
{"title":"Note on C*-algebras associated to boundary actions of hyperbolic 3-manifold groups","authors":"Shirly Geffen, Julian Kranz","doi":"arxiv-2407.15215","DOIUrl":"https://doi.org/arxiv-2407.15215","url":null,"abstract":"Using Kirchberg-Phillips' classification of purely infinite C*-algebras by\u0000K-theory, we prove that the isomorphism types of crossed product C*-algebras\u0000associated to certain hyperbolic 3-manifold groups acting on their Gromov\u0000boundary only depend on the manifold's homology. As a result, we obtain\u0000infinitely many pairwise non-isomorphic hyperbolic groups all of whose\u0000associated crossed products are isomorphic. These isomomorphisms are not of\u0000dynamical nature in the sense that they are not induced by isomorphisms of the\u0000underlying groupoids.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784060","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}
For a complex Banach space $mathbb X$, we prove that $mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $mathbb X$ the function $A_T: mathbb X rightarrow [0, infty]$ defined by $A_T(x)=(|x|^2 -|Tx|^2)^{frac{1}{2}}$ gives a norm on $mathbb X$. We also find several other necessary and sufficient conditions in this thread such that a Banach sapce becomes a Hilbert space. We construct examples of strict contractions on non-Hilbert Banach spaces that do not dilate to isometries. Then we characterize all strict contractions on a non-Hilbert Banach space that dilate to isometries and find explicit isometric dilation for them. We prove several other results including characterizations of complemented subspaces in a Banach space, extension of a Wold isometry to a Banach space unitary and describing norm attainment sets of Banach space operators in terms of dilations.
{"title":"A dilation theoretic approach to Banach spaces","authors":"Swapan Jana, Sourav Pal, Saikat Roy","doi":"arxiv-2407.15112","DOIUrl":"https://doi.org/arxiv-2407.15112","url":null,"abstract":"For a complex Banach space $mathbb X$, we prove that $mathbb X$ is a\u0000Hilbert space if and only if every strict contraction $T$ on $mathbb X$\u0000dilates to an isometry if and only if for every strict contraction $T$ on\u0000$mathbb X$ the function $A_T: mathbb X rightarrow [0, infty]$ defined by\u0000$A_T(x)=(|x|^2 -|Tx|^2)^{frac{1}{2}}$ gives a norm on $mathbb X$. We also\u0000find several other necessary and sufficient conditions in this thread such that\u0000a Banach sapce becomes a Hilbert space. We construct examples of strict\u0000contractions on non-Hilbert Banach spaces that do not dilate to isometries.\u0000Then we characterize all strict contractions on a non-Hilbert Banach space that\u0000dilate to isometries and find explicit isometric dilation for them. We prove\u0000several other results including characterizations of complemented subspaces in\u0000a Banach space, extension of a Wold isometry to a Banach space unitary and\u0000describing norm attainment sets of Banach space operators in terms of\u0000dilations.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786100","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 focuses on the cohomology of operator algebras associated with the free semigroup generated by the set ${z_{alpha}}_{alphainLambda}$, with the left regular free semigroup algebra $mathfrak{L}_{Lambda}$ and the non-commutative disc algebra $mathfrak{A}_{Lambda}$ serving as two typical examples. We establish that all derivations of these algebras are automatically continuous. By introducing a novel computational approach, we demonstrate that the first Hochschild cohomology group of $mathfrak{A}_{Lambda}$ with coefficients in $mathfrak{L}_{Lambda}$ is zero. Utilizing the Ces`aro operators and conditional expectations, we show that the first normal cohomology group of $mathfrak{L}_{Lambda}$ is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when $|Lambda|
{"title":"Hochschild cohomology for free semigroup algebras","authors":"Linzhe Huang, Minghui Ma, Xiaomin Wei","doi":"arxiv-2407.14729","DOIUrl":"https://doi.org/arxiv-2407.14729","url":null,"abstract":"This paper focuses on the cohomology of operator algebras associated with the\u0000free semigroup generated by the set ${z_{alpha}}_{alphainLambda}$, with\u0000the left regular free semigroup algebra $mathfrak{L}_{Lambda}$ and the\u0000non-commutative disc algebra $mathfrak{A}_{Lambda}$ serving as two typical\u0000examples. We establish that all derivations of these algebras are automatically\u0000continuous. By introducing a novel computational approach, we demonstrate that\u0000the first Hochschild cohomology group of $mathfrak{A}_{Lambda}$ with\u0000coefficients in $mathfrak{L}_{Lambda}$ is zero. Utilizing the Ces`aro\u0000operators and conditional expectations, we show that the first normal\u0000cohomology group of $mathfrak{L}_{Lambda}$ is trivial. Finally, we prove that\u0000the higher cohomology groups of the non-commutative disc algebras with\u0000coefficients in the complex field vanish when $|Lambda|<infty$. These methods\u0000extend to compute the cohomology groups of a specific class of operator\u0000algebras generated by the left regular representations of cancellative\u0000semigroups, which notably include Thompson's semigroup.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784128","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 introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek cite{Lambek} and Johnstone cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 le k le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties which are impossible for $1$-trees. Finally, we collect more examples of higher-rank planar trees which are not in our family.
我们引入了一个新的高阶图族，其构造受到了兰姆贝克（Lambek）和约翰斯通（Johnstone）用于单类和类嵌入结果的图形技术的启发。我们证明了它们是 2 ~ k ~ 4$ 的平面 $k$ 树。我们还举例说明了高阶树与$1$树的区别，因为高阶树具有$1$树不可能具有的性质。最后，我们收集了更多不属于我们家族的高阶平面树的例子。
{"title":"Higher-rank trees: Finite higher-rank planar trees arising from polyhedral graphs. Differences from one-trees","authors":"David Pask","doi":"arxiv-2407.14048","DOIUrl":"https://doi.org/arxiv-2407.14048","url":null,"abstract":"We introduce a new family of higher-rank graphs, whose construction was\u0000inspired by the graphical techniques of Lambek cite{Lambek} and Johnstone\u0000cite{Johnstone} used for monoid and category emedding results. We show that\u0000they are planar $k$-trees for $2 le k le 4$. We also show that higher-rank\u0000trees differ from $1$-trees by giving examples of higher-rank trees having\u0000properties which are impossible for $1$-trees. Finally, we collect more\u0000examples of higher-rank planar trees which are not in our family.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740681","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}
With the aim of understanding the localization topology correspondence for non periodic gapped quantum systems, we investigate the relation between the existence of an algebraically well-localized generalized Wannier basis and the topological triviality of the corresponding projection operator. Inspired by the work of M. Ludewig and G.C. Thiang, we consider the triviality of a projection in the sense of coarse geometry, i.e. as triviality in the $K_0$-theory of the Roe $C^*$-algebra of $mathrm{R}^d$. We obtain in Theorem 2.8 a threshold, depending on the dimension, for the decay rate of the generalized Wannier functions which implies topological triviality in Roe sense. This threshold reduces, for $d = 2$, to the almost optimal threshold appearing in the Localization Dichotomy Conjecture.
{"title":"Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension","authors":"Vincenzo Rossi, Gianluca Panati","doi":"arxiv-2407.14235","DOIUrl":"https://doi.org/arxiv-2407.14235","url":null,"abstract":"With the aim of understanding the localization topology correspondence for\u0000non periodic gapped quantum systems, we investigate the relation between the\u0000existence of an algebraically well-localized generalized Wannier basis and the\u0000topological triviality of the corresponding projection operator. Inspired by\u0000the work of M. Ludewig and G.C. Thiang, we consider the triviality of a\u0000projection in the sense of coarse geometry, i.e. as triviality in the\u0000$K_0$-theory of the Roe $C^*$-algebra of $mathrm{R}^d$. We obtain in Theorem\u00002.8 a threshold, depending on the dimension, for the decay rate of the\u0000generalized Wannier functions which implies topological triviality in Roe\u0000sense. This threshold reduces, for $d = 2$, to the almost optimal threshold\u0000appearing in the Localization Dichotomy Conjecture.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745947","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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