Canadian Journal of Mathematics最新文献

{"title":"Linear homeomorphisms of function spaces and the position of a space in its compactification","authors":"Mikołaj Krupski","doi":"10.4153/s0008414x23000779","DOIUrl":"https://doi.org/10.4153/s0008414x23000779","url":null,"abstract":"<p>An old question of Arhangel’skii asks if the Menger property of a Tychonoff space <span>X</span> is preserved by homeomorphisms of the space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220142853673-0028:S0008414X23000779:S0008414X23000779_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$C_p(X)$</span></span></img></span></span> of continuous real-valued functions on <span>X</span> endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220142853673-0028:S0008414X23000779:S0008414X23000779_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$C_p(X)$</span></span></img></span></span> by looking at the way <span>X</span> is positioned in its (Čech–Stone) compactification.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138824009","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}

{"title":"More Ramsey theory for highly connected monochromatic subgraphs","authors":"Michael Hrušák, Saharon Shelah, Jing Zhang","doi":"10.4153/s0008414x23000767","DOIUrl":"https://doi.org/10.4153/s0008414x23000767","url":null,"abstract":"<p>An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, <span>Acta Mathematica Hungarica</span> 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC: </p><ul><li><p><span>•</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104113709561-0705:S0008414X23000767:S0008414X23000767_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$kappa to _{hc} (kappa )^2_omega $</span></span></img></span></span> for every regular cardinal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104113709561-0705:S0008414X23000767:S0008414X23000767_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$kappa geq aleph _2$</span></span></img></span></span>,</p></li><li><p><span>•</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104113709561-0705:S0008414X23000767:S0008414X23000767_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$neg mathsf {CH}+ aleph _2 to _{hc} (aleph _1)^2_omega $</span></span></img></span></span>.</p></li></ul><p></p><p>Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that </p><ul><li><p><span>•</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104113709561-0705:S0008414X23000767:S0008414X23000767_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$aleph _2 to _{hc} [aleph _2]^2_{omega ,2}$</span></span></img></span></span> is consistent with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104113709561-0705:S0008414X23000767:S0008414X23000767_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$neg mathsf {CH}$</span></span></img></span></span>.</p></li></ul><p></p><p>To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105583","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}

{"title":"Linear maps preserving (p, k) norms of tensor products of matrices","authors":"Zejun Huang, Nung-Sing Sze, Run Zheng","doi":"10.4153/s0008414x23000858","DOIUrl":"https://doi.org/10.4153/s0008414x23000858","url":null,"abstract":"Let $m,nge 2$ be integers. Denote by $M_n$ the set of $ntimes n$ complex matrices. Let $|cdot|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1leq kleq mn$ and $2<p<infty$. We show that a linear map $phi:M_{mn}rightarrow M_{mn}$ satisfies $$|phi(Aotimes B)|_{(p,k)}=|Aotimes B|_{(p,k)} {rmquad for~ allquad}Ain M_m {rm ~and ~}Bin M_n$$ if and only if there exist unitary matrices $U,Vin M_{mn}$ such that $$phi(Aotimes B)=U(varphi_1(A)otimes varphi_2(B))V {rmquad for~ allquad}Ain M_m {rm~ and~ }Bin M_n,$$ where $varphi_s$ is the identity map or the transposition map $Xto X^T$ for $s=1,2$. The result is also extended to multipartite systems.","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139365709","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}

{"title":"SYMMETRIC AND ANTISYMMETRIC TENSOR PRODUCTS FOR THE FUNCTION-THEORETIC OPERATOR THEORIST","authors":"S. Garcia, Ryan O’Loughlin, Jiahui Yu","doi":"10.4153/s0008414x23000901","DOIUrl":"https://doi.org/10.4153/s0008414x23000901","url":null,"abstract":"We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139368487","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}