In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $lambda$-Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This $lambda$-Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal $lambda$-Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open questions raised by Gromov. As another application, we prove Gromov's $square^{n-m}$ inequality on the bound of distances between opposite faces of spin manifolds with cube-like boundaries with a suboptimal constant. As immediate consequences, this implies Gromov's cube inequality on the bound of widths of Riemannian cubes and Gromov's conjecture on the bound of widths of Riemannian bands with suboptimal constants. Further geometric applications will be discussed in a forthcoming paper.
{"title":"A quantitative relative index theorem and Gromov's conjectures on positive scalar curvature","authors":"Zhizhang Xie","doi":"10.4171/jncg/504","DOIUrl":"https://doi.org/10.4171/jncg/504","url":null,"abstract":"In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $lambda$-Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This $lambda$-Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal $lambda$-Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open questions raised by Gromov. As another application, we prove Gromov's $square^{n-m}$ inequality on the bound of distances between opposite faces of spin manifolds with cube-like boundaries with a suboptimal constant. As immediate consequences, this implies Gromov's cube inequality on the bound of widths of Riemannian cubes and Gromov's conjecture on the bound of widths of Riemannian bands with suboptimal constants. Further geometric applications will be discussed in a forthcoming paper.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42087619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we give a new class of automorphisms of Leavitt path algebras of arbitrary graphs. Consequently, we obtain Anick type automorphisms of these Leavitt path algebras and new irreducible representations of Leavitt algebras of type $(1, n)$.
{"title":"Anick type automorphisms and new irreducible representations of Leavitt path algebras","authors":"S. Kuroda, T. G. Nam","doi":"10.4171/jncg/489","DOIUrl":"https://doi.org/10.4171/jncg/489","url":null,"abstract":"In this article, we give a new class of automorphisms of Leavitt path algebras of arbitrary graphs. Consequently, we obtain Anick type automorphisms of these Leavitt path algebras and new irreducible representations of Leavitt algebras of type $(1, n)$.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41805085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.
{"title":"Calabi–Yau structures for multiplicative preprojective algebras","authors":"T. Bozec, D. Calaque, Sarah Scherotzke","doi":"10.4171/JNCG/488","DOIUrl":"https://doi.org/10.4171/JNCG/488","url":null,"abstract":"In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41658277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new proof of the non-triviality of wheel graph homology classes using higher operations on Lie graph homology and a derived version of Koszul duality for modular operads.
{"title":"Wheel graph homology classes via Lie graph homology","authors":"Benjamin C. Ward","doi":"10.4171/jncg/508","DOIUrl":"https://doi.org/10.4171/jncg/508","url":null,"abstract":"We give a new proof of the non-triviality of wheel graph homology classes using higher operations on Lie graph homology and a derived version of Koszul duality for modular operads.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49089889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an action of a finite cyclic group $F$ on an $n$-dimensional noncommutative torus $A_theta,$ we give sufficient conditions when the fundamental projective modules over $A_theta$, which determine the range of the canonical trace on $A_theta,$ extend to projective modules over the crossed product C*-algebra $A_theta rtimes F.$ Our results allow us to understand the range of the canonical trace on $A_theta rtimes F$, and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of $mathbb{Z}_2$ on any $n$-dimensional noncommutative torus. As an application, for the flip action of $mathbb{Z}_2$ on a simple $n$-dimensional torus $A_theta$, we determine the Morita equivalence class of $A_theta rtimes mathbb{Z}_2,$ in terms of the Morita equivalence class of $A_theta.$
{"title":"Tracing projective modules over noncommutative orbifolds","authors":"Sayan Chakraborty","doi":"10.4171/jncg/487","DOIUrl":"https://doi.org/10.4171/jncg/487","url":null,"abstract":"For an action of a finite cyclic group $F$ on an $n$-dimensional noncommutative torus $A_theta,$ we give sufficient conditions when the fundamental projective modules over $A_theta$, which determine the range of the canonical trace on $A_theta,$ extend to projective modules over the crossed product C*-algebra $A_theta rtimes F.$ Our results allow us to understand the range of the canonical trace on $A_theta rtimes F$, and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of $mathbb{Z}_2$ on any $n$-dimensional noncommutative torus. As an application, for the flip action of $mathbb{Z}_2$ on a simple $n$-dimensional torus $A_theta$, we determine the Morita equivalence class of $A_theta rtimes mathbb{Z}_2,$ in terms of the Morita equivalence class of $A_theta.$","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47803754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the K-theory of twisted groupoid C*-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum–Connes conjecture with coefficients gives rise to an isomorphism between the K-theory groups of the respective twisted groupoid C*-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault–Deaconu groupoids and P-graph algebras.
{"title":"K-theory and homotopies of twists on ample groupoids","authors":"Christian Bönicke","doi":"10.4171/JNCG/399","DOIUrl":"https://doi.org/10.4171/JNCG/399","url":null,"abstract":"This paper investigates the K-theory of twisted groupoid C*-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum–Connes conjecture with coefficients gives rise to an isomorphism between the K-theory groups of the respective twisted groupoid C*-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault–Deaconu groupoids and P-graph algebras.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42495650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper provides a version of the rational Hodge conjecture for $3dg$ categories. The noncommutative Hodge conjecture is equivalent to the version proposed in cite{perry2020integral} for admissible subcategories. We obtain examples of evidence of the Hodge conjecture by techniques of noncommutative geometry. Finally, we show that the noncommutative Hodge conjecture for smooth proper connective $3dg$ algebras is true.
{"title":"Noncommutative Hodge conjecture","authors":"Xun Lin","doi":"10.4171/jncg/517","DOIUrl":"https://doi.org/10.4171/jncg/517","url":null,"abstract":"The paper provides a version of the rational Hodge conjecture for $3dg$ categories. The noncommutative Hodge conjecture is equivalent to the version proposed in cite{perry2020integral} for admissible subcategories. We obtain examples of evidence of the Hodge conjecture by techniques of noncommutative geometry. Finally, we show that the noncommutative Hodge conjecture for smooth proper connective $3dg$ algebras is true.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41466174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider various statements that characterize the hyperfinite II$_1$ factors amongst embeddable II$_1$ factors in the non-embeddable situation. In particular, we show that"generically"a II$_1$ factor has the Jung property (which states that every embedding of itself into its ultrapower is unitarily conjugate to the diagonal embedding) if and only if it is self-tracially stable (which says that every such embedding has an approximate lifting). We prove that the enforceable factor, should it exist, has these equivalent properties. Our techniques are model-theoretic in nature. We also show how these techniques can be used to give new proofs that the hyperfinite II$_1$ factor has the aforementioned properties.
{"title":"Non-embeddable II$_1$ factors resembling the hyperfinite II$_1$ factor","authors":"Isaac Goldbring","doi":"10.4171/jncg/474","DOIUrl":"https://doi.org/10.4171/jncg/474","url":null,"abstract":"We consider various statements that characterize the hyperfinite II$_1$ factors amongst embeddable II$_1$ factors in the non-embeddable situation. In particular, we show that\"generically\"a II$_1$ factor has the Jung property (which states that every embedding of itself into its ultrapower is unitarily conjugate to the diagonal embedding) if and only if it is self-tracially stable (which says that every such embedding has an approximate lifting). We prove that the enforceable factor, should it exist, has these equivalent properties. Our techniques are model-theoretic in nature. We also show how these techniques can be used to give new proofs that the hyperfinite II$_1$ factor has the aforementioned properties.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48147732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the philosophy that C∗-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of C∗-algebras. We focus on C∗-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable ∞-category of noncommutative CW-spectra, which we denote by NSp. Let M be the full spectral subcategory of NSp spanned by “noncommutative suspension spectra” of matrix algebras. Our main result is that NSp is equivalent to the ∞-category of spectral presheaves on M. To prove this we first prove a general result which states that any compactly generated stable ∞-category is naturally equivalent to the ∞category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an ∞-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched ∞-categories as developed by Hinich [Hin2, Hin3]. We end by presenting a “strict” model for M. That is, we define a category Ms strictly enriched in a certain monoidal model category of spectra Sp. We give a direct proof that the category of Sp-enriched presheaves Mop s → Sp M with the projective model structure models NSp and conclude that Ms is a strict model for M.
{"title":"Noncommutative CW-spectra as enriched presheaves on matrix algebras","authors":"G. Arone, Ilan Barnea, T. Schlank","doi":"10.4171/jncg/481","DOIUrl":"https://doi.org/10.4171/jncg/481","url":null,"abstract":"Motivated by the philosophy that C∗-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of C∗-algebras. We focus on C∗-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable ∞-category of noncommutative CW-spectra, which we denote by NSp. Let M be the full spectral subcategory of NSp spanned by “noncommutative suspension spectra” of matrix algebras. Our main result is that NSp is equivalent to the ∞-category of spectral presheaves on M. To prove this we first prove a general result which states that any compactly generated stable ∞-category is naturally equivalent to the ∞category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an ∞-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched ∞-categories as developed by Hinich [Hin2, Hin3]. We end by presenting a “strict” model for M. That is, we define a category Ms strictly enriched in a certain monoidal model category of spectra Sp. We give a direct proof that the category of Sp-enriched presheaves Mop s → Sp M with the projective model structure models NSp and conclude that Ms is a strict model for M.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45178280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. Biswas, Saikat Chatterjee, Praphulla Koushik, F. Neumann
We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.
利用与横向切向分布相关的Atiyah序列，构造并研究了李群和可微堆及其上的主束上的一般连接。
{"title":"Atiyah sequences and connections on principal bundles over Lie groupoids and differentiable stacks","authors":"I. Biswas, Saikat Chatterjee, Praphulla Koushik, F. Neumann","doi":"10.4171/jncg/486","DOIUrl":"https://doi.org/10.4171/jncg/486","url":null,"abstract":"We construct and study general connections on Lie groupoids and differentiable stacks as well as on principal bundles over them using Atiyah sequences associated to transversal tangential distributions.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44970331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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