{"title":"Unitary groups, -theory, and traces","authors":"Pawel Sarkowicz","doi":"10.1017/s0017089523000447","DOIUrl":"https://doi.org/10.1017/s0017089523000447","url":null,"abstract":"We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089523000447_inline2.png\" /> <jats:tex-math> $K$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-theoretic regularity conditions, these maps can be seen to commute with the pairing between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089523000447_inline3.png\" /> <jats:tex-math> $K_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138691449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The connective","authors":"Donald M. Davis, W. Stephen Wilson","doi":"10.1017/s0017089523000423","DOIUrl":"https://doi.org/10.1017/s0017089523000423","url":null,"abstract":"<p>We compute <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$ku^*left(K!left({mathbb{Z}}_p,2right)right)$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$ku_*left(K!left({mathbb{Z}}_p,2right)right)$</span></span></img></span></span>, the connective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$KU$</span></span></img></span></span>-cohomology and connective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$KU$</span></span></img></span></span>-homology groups of the mod-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> Eilenberg–MacLane space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$K!left({mathbb{Z}}_p,2right)$</span></span></img></span></span>, using the Adams spectral sequence. We obtain a striking interaction between <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$h_0$</span></span></img></span></span>-extensions and exotic extensions. The mod-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> connective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$KU$</span></span></img></span></span>-cohomology groups, computed elsewhere, are needed in order to establish higher differentials a","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138569427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on almost Yamabe solitons","authors":"Wagner Oliveira Costa-Filho","doi":"10.1017/s0017089523000411","DOIUrl":"https://doi.org/10.1017/s0017089523000411","url":null,"abstract":"In this paper, we present a sufficient condition for almost Yamabe solitons to have constant scalar curvature. Additionally, under some geometric scenarios, we provide some triviality and rigidity results for these structures.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On -unramified extensions over imaginary quadratic fields","authors":"Kwang-Seob Kim, Joachim König","doi":"10.1017/s001708952300037x","DOIUrl":"https://doi.org/10.1017/s001708952300037x","url":null,"abstract":"<jats:p>Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300037X_inline2.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an integer congruent to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300037X_inline3.png\" /> <jats:tex-math> $0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300037X_inline4.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> modulo <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300037X_inline5.png\" /> <jats:tex-math> $4$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Under the assumption of the ABC conjecture, we prove that, given any integer <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300037X_inline6.png\" /> <jats:tex-math> $m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300037X_inline7.png\" /> <jats:tex-math> $A_n times C_m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. The same result is obtained unconditionally in special cases.</jats:p>","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A classification of some thick subcategories in locally noetherian Grothendieck categories","authors":"Kaili Wu, Xinchao Ma","doi":"10.1017/s001708952300040x","DOIUrl":"https://doi.org/10.1017/s001708952300040x","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300040X_inline1.png\" /> <jats:tex-math> $mathcal{A}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a locally noetherian Grothendieck category. We classify all full subcategories of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S001708952300040X_inline2.png\" /> <jats:tex-math> $mathcal{A}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> which are thick and closed under taking arbitrary direct sums and injective envelopes by injective spectrum. This result gives a generalization from the commutative noetherian ring to the locally noetherian Grothendieck category.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A natural pseudometric on homotopy groups of metric spaces","authors":"Jeremy Brazas, Paul Fabel","doi":"10.1017/s0017089523000393","DOIUrl":"https://doi.org/10.1017/s0017089523000393","url":null,"abstract":"Abstract For a path-connected metric space $(X,d)$ , the $n$ -th homotopy group $pi _n(X)$ inherits a natural pseudometric from the $n$ -th iterated loop space with the uniform metric. This pseudometric gives $pi _n(X)$ the structure of a topological group, and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$ . In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $pi _n(X)$ . Our main result is that the pseudometric topology agrees with the shape topology on $pi _n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135390259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paul Alexander Helminck, Yassine El Maazouz, Enis Kaya
{"title":"Tropical invariants for binary quintics and reduction types of Picard curves","authors":"Paul Alexander Helminck, Yassine El Maazouz, Enis Kaya","doi":"10.1017/s0017089523000344","DOIUrl":"https://doi.org/10.1017/s0017089523000344","url":null,"abstract":"Abstract In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $overline{M}_{0,n}$ .","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135636939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Notes on hyperelliptic mapping class groups","authors":"Marco Boggi","doi":"10.1017/s0017089523000381","DOIUrl":"https://doi.org/10.1017/s0017089523000381","url":null,"abstract":"Abstract Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural epimorphisms between mapping class groups of surfaces with marked points. We study these groups in a systematic way. An application of this theory is a counterexample to the genus $2$ case of a conjecture by Putman and Wieland on virtual linear representations of mapping class groups. In the last section, we study profinite completions of hyperelliptic mapping class groups: we extend the congruence subgroup property to the general class of hyperelliptic mapping class groups introduced above and then determine the centralizers of multitwists and of open subgroups in their profinite completions.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the reducing projective dimension over local rings","authors":"Olgur Celikbas, Souvik Dey, Toshinori Kobayashi, Hiroki Matsui","doi":"10.1017/s0017089523000368","DOIUrl":"https://doi.org/10.1017/s0017089523000368","url":null,"abstract":"Abstract In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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