We study embeddings of uniform Roe algebras which have "large range" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding $Phi:mathrm{C}^*_u(X)to mathrm{C}^*_u(Y)$ with "large range" and so that $Phi(ell_infty(X))$ is a Cartan subalgebra of $mathrm{C}^*_u(Y)$, then there is a bijective coarse quotient $Xto Y$. This shows that the large scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For instance, if $X$ has finite asymptotic dimension, so does $Y$.
{"title":"Coarse quotients of metric spaces and embeddings of uniform Roe algebras","authors":"B. M. Braga","doi":"10.4171/jncg/463","DOIUrl":"https://doi.org/10.4171/jncg/463","url":null,"abstract":"We study embeddings of uniform Roe algebras which have \"large range\" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding $Phi:mathrm{C}^*_u(X)to mathrm{C}^*_u(Y)$ with \"large range\" and so that $Phi(ell_infty(X))$ is a Cartan subalgebra of $mathrm{C}^*_u(Y)$, then there is a bijective coarse quotient $Xto Y$. This shows that the large scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For instance, if $X$ has finite asymptotic dimension, so does $Y$.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43239191","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 prove a number of results having to do with equipping type-I $mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the $mathrm{C}^*$-algebra in question is an extension of a non-zero finite direct sum of elementary $mathrm{C}^*$-algebras by a commutative unital $mathrm{C}^*$-algebra then it must be finite-dimensional.
{"title":"Compact quantum group structures on type-I $mathrm{C}^*$-algebras","authors":"A. Chirvasitu, Jacek Krajczok, P. Sołtan","doi":"10.4171/jncg/516","DOIUrl":"https://doi.org/10.4171/jncg/516","url":null,"abstract":"We prove a number of results having to do with equipping type-I $mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the $mathrm{C}^*$-algebra in question is an extension of a non-zero finite direct sum of elementary $mathrm{C}^*$-algebras by a commutative unital $mathrm{C}^*$-algebra then it must be finite-dimensional.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45446624","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 prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved L ∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for L ∞ algebras. The main ingredients are the obstruction theory for L ∞ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved L ∞ algebras. Both techniques work in the A ∞ case as well.
{"title":"The inverse function theorem for curved $L$-infinity spaces","authors":"Lino Amorim, Junwu Tu","doi":"10.4171/jncg/484","DOIUrl":"https://doi.org/10.4171/jncg/484","url":null,"abstract":". In this paper we prove an inverse function theorem in derived differential geometry. More concretely, we show that a morphism of curved L ∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. This theorem simultaneously generalizes the inverse function theorem for smooth manifolds and the Whitehead theorem for L ∞ algebras. The main ingredients are the obstruction theory for L ∞ homomorphisms (in the curved setting) and the homotopy transfer theorem for curved L ∞ algebras. Both techniques work in the A ∞ case as well.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45040913","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 prove that if $Gamma$ is a group of polynomial growth then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocyle we thus define a higher analogue of Lott's delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if $Gamma$ is of polynomial growth then there is a well defined pairing between delocalized cyclic cocyles and $K$-theory classes of $C^*$-algebraic secondary higher invariants. When this $K$-theory class is that of a higher rho invariant of an invertible differential operator we show this pairing is precisely the aforementioned higher analogue of Lott's delocalized eta invariant. As an application of this equivalence we provide a delocalized higher Atiyah-Patodi-Singer index theorem given $M$ is a compact spin manifold with boundary, equipped with a positive scalar metric $g$ and having fundamental group $Gamma=pi_1(M)$ which is finitely generated and of polynomial growth.
{"title":"Secondary higher invariants and cyclic cohomology for groups of polynomial growth","authors":"Sheagan A. K. A. John","doi":"10.4171/jncg/456","DOIUrl":"https://doi.org/10.4171/jncg/456","url":null,"abstract":"We prove that if $Gamma$ is a group of polynomial growth then each delocalized cyclic cocycle on the group algebra has a representative of polynomial growth. For each delocalized cocyle we thus define a higher analogue of Lott's delocalized eta invariant and prove its convergence for invertible differential operators. We also use a determinant map construction of Xie and Yu to prove that if $Gamma$ is of polynomial growth then there is a well defined pairing between delocalized cyclic cocyles and $K$-theory classes of $C^*$-algebraic secondary higher invariants. When this $K$-theory class is that of a higher rho invariant of an invertible differential operator we show this pairing is precisely the aforementioned higher analogue of Lott's delocalized eta invariant. As an application of this equivalence we provide a delocalized higher Atiyah-Patodi-Singer index theorem given $M$ is a compact spin manifold with boundary, equipped with a positive scalar metric $g$ and having fundamental group $Gamma=pi_1(M)$ which is finitely generated and of polynomial growth.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43770415","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}
{"title":"Embedding of the derived Brauer group into the secondary $K$-theory ring","authors":"Gonçalo Tabuada","doi":"10.4171/jncg/379","DOIUrl":"https://doi.org/10.4171/jncg/379","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"14 1","pages":"773-788"},"PeriodicalIF":0.9,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48348501","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 study the covariant derivatives of an eigenfunction for the Laplace-Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.
{"title":"Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure","authors":"Fei Qi","doi":"10.4171/jncg/472","DOIUrl":"https://doi.org/10.4171/jncg/472","url":null,"abstract":"We study the covariant derivatives of an eigenfunction for the Laplace-Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46679387","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 show that every separable C*-algebra of real rank zero that tensorially absorbs the Jiang-Su algebra contains a dense set of generators. It follows that in every classifiable, simple, nuclear C*-algebra, a generic element is a generator.
{"title":"Generators in $mathcal{Z}$-stable $C^*$-algebras of real rank zero","authors":"Hannes Thiel","doi":"10.4171/jncg/454","DOIUrl":"https://doi.org/10.4171/jncg/454","url":null,"abstract":"We show that every separable C*-algebra of real rank zero that tensorially absorbs the Jiang-Su algebra contains a dense set of generators. \u0000It follows that in every classifiable, simple, nuclear C*-algebra, a generic element is a generator.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46401755","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 [14], Weinberger, Xie and Yu proved that higher rho invariant associated to homotopy equivalence defines a group homomorphism from the topological structure group to analytic structure group, K-theory of certain geometric C∗-algebras, by piecewise-linear approach. In this paper, we adapt part of Weinberger, Xie and Yu’s work, to give a differential geometry theoretic proof of the additivity of the map induced by higher rho invariant associated to homotopy equivalence on topological structure group. Mathematics Subject Classification (2010). 58J22.
{"title":"Additivity of higher rho invariant for the topological structure group from a differential point of view","authors":"Baojie Jiang, Hongzhi Liu","doi":"10.4171/jncg/369","DOIUrl":"https://doi.org/10.4171/jncg/369","url":null,"abstract":"In [14], Weinberger, Xie and Yu proved that higher rho invariant associated to homotopy equivalence defines a group homomorphism from the topological structure group to analytic structure group, K-theory of certain geometric C∗-algebras, by piecewise-linear approach. In this paper, we adapt part of Weinberger, Xie and Yu’s work, to give a differential geometry theoretic proof of the additivity of the map induced by higher rho invariant associated to homotopy equivalence on topological structure group. Mathematics Subject Classification (2010). 58J22.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"14 1","pages":"441-486"},"PeriodicalIF":0.9,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47362788","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}
After embedding the objects quasifolds into the category {Diffeology}, we associate a C*-agebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry (beginning with the today classical example of the irrational torus) which associates a Morita class of C*-algebras with a diffeomorphic class of quasifolds.
{"title":"Quasifolds, diffeology and noncommutative geometry","authors":"Patrick Iglesias-Zemmour, E. Prato","doi":"10.4171/JNCG/419","DOIUrl":"https://doi.org/10.4171/JNCG/419","url":null,"abstract":"After embedding the objects quasifolds into the category {Diffeology}, we associate a C*-agebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry (beginning with the today classical example of the irrational torus) which associates a Morita class of C*-algebras with a diffeomorphic class of quasifolds.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44191272","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 generalise a number of classical results from the theory of KMS states to KMS weights in the setting of $C^{*}$-dynamical systems arising from a continuous groupoid homomorphism $c:mathcal{G} to mathbb{R}$ on a locally compact second countable Hausdorff etale groupoid $mathcal{G}$. In particular, we generalise Neshveyev's Theorem to KMS weights.
我们将KMS态理论的一些经典结果推广到由局部紧第二可数Hausdorff etale群胚$mathcal｛G｝$上的连续群胚同态$C:mathcal｛G} to mathbb｛R｝$引起的$C^{*}$动力系统设置中的KMS权。特别地，我们将Neshveyev定理推广到KMS权。
{"title":"The structure of KMS weights on étale groupoid C*-algebras","authors":"J. Christensen","doi":"10.4171/jncg/507","DOIUrl":"https://doi.org/10.4171/jncg/507","url":null,"abstract":"We generalise a number of classical results from the theory of KMS states to KMS weights in the setting of $C^{*}$-dynamical systems arising from a continuous groupoid homomorphism $c:mathcal{G} to mathbb{R}$ on a locally compact second countable Hausdorff etale groupoid $mathcal{G}$. In particular, we generalise Neshveyev's Theorem to KMS weights.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45810797","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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