Proceedings of the American Mathematical Society, Series B最新文献

{"title":"On the divergence of Taylor series in de Branges–Rovnyak spaces","authors":"Pierre-Olivier Parisé, Thomas Ransford","doi":"10.1090/bproc/176","DOIUrl":"https://doi.org/10.1090/bproc/176","url":null,"abstract":"It is known that there exist functions in certain de Branges–Rovnyak spaces whose Taylor series diverge in norm, even though polynomials are dense in the space. This is often proved by showing that the sequence of Taylor partial sums is unbounded in norm. In this note we show that it can even happen that the Taylor partial sums tend to infinity in norm. We also establish similar results for lower-triangular summability methods such as the Cesàro means.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"70 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140973883","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":"A note on the anisotropic Bernstein problem in ℝ³","authors":"César Rosales","doi":"10.1090/bproc/214","DOIUrl":"https://doi.org/10.1090/bproc/214","url":null,"abstract":"It was proved by Jenkins [Arch. Rational Mech. Anal. 8 (1961), 181–206] that a smooth entire graph in \u0000\u0000 \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 3\u0000 \u0000 {mathbb {R}}^3\u0000 \u0000\u0000 with vanishing anisotropic mean curvature must be a plane. By using a calibration argument and a stability inequality we show here a different self-contained proof of this result, which is still valid when the anisotropic mean curvature is constant.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"28 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140972786","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":"Corrigendum to “intermediate C*-algebras of Cartan embeddings”","authors":"Jonathan Brown, Ruy Exel, Adam Fuller, David Pitts, Sarah Reznikoff","doi":"10.1090/bproc/169","DOIUrl":"https://doi.org/10.1090/bproc/169","url":null,"abstract":"","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"4 S3B","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140725265","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":"Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces","authors":"N. Zorboska","doi":"10.1090/bproc/215","DOIUrl":"https://doi.org/10.1090/bproc/215","url":null,"abstract":"<p>For a complex Borel measure <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\u0000 <mml:semantics>\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">mu</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> on the open unit disk, and for a weighted Dirichlet space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H Subscript s\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>s</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {H}_s</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than s greater-than 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">0>s>1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we characterize the boundedness of the measure induced Hankel type operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript mu comma s Baseline colon script upper H Subscript s Baseline right-arrow script upper H Subscript s Baseline overbar\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>s</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mover>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">H</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>s</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H_{mu ,s}: mathcal {H}_s to overline {mathcal {H}_s}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H squared equals script upper H 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mn>2","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140726529","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":"Regularity of sections of CR vector bundles","authors":"B. Lamel, N. Mir","doi":"10.1090/bproc/149","DOIUrl":"https://doi.org/10.1090/bproc/149","url":null,"abstract":"<p>In this note, we show that every generalized section <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\">\u0000 <mml:semantics>\u0000 <mml:mi>σ<!-- σ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of a CR vector bundle <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\u0000 <mml:semantics>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over a CR manifold <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has the property that near most points of its singular support, there exists a proper abstract CR subbundle <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F subset-of upper E\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:mi>E</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">F subset E</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which has the property that every <italic>real</italic> subbundle of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\">\u0000 <mml:semantics>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">E</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> which contains the image of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma\">\u0000 <mml:semantics>\u0000 <mml:mi>σ<!-- σ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">sigma</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> also contains <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"33 S1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140265188","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":"𝐾-theory of multiparameter persistence modules: Additivity","authors":"Ryan E. Grady, Anna Schenfisch","doi":"10.1090/bproc/208","DOIUrl":"https://doi.org/10.1090/bproc/208","url":null,"abstract":"Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group \u0000\u0000 \u0000 \u0000 K\u0000 0\u0000 \u0000 K_0\u0000 \u0000\u0000 with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"128 19","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140078642","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":"The stable exotic Cuntz algebras are higher-rank graph algebras","authors":"Jeffrey L. Boersema, Sarah Browne, Elizabeth Gillaspy","doi":"10.1090/bproc/180","DOIUrl":"https://doi.org/10.1090/bproc/180","url":null,"abstract":"<p>For each odd integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n geq 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we construct a rank-3 graph <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">Lambda _n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with involution <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">gamma _n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> whose real <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript double-struck upper R Superscript asterisk Baseline left-parenthesis normal upper Lamda Subscript n Baseline comma gamma Subscript n Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mstyle displaystyle=\"false\" scriptlevel=\"2\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mstyle>\u0000 </mml:mrow>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">C^*_{scriptscriptstyle mathbb {R}}(Lambda _n, gamma _n)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is stably isomorphic to the exotic Cuntz algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"scr","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"118 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140079458","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":"The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems","authors":"Eva Belmont, Zhouli Xu, Shangjie Zhang","doi":"10.1090/bproc/203","DOIUrl":"https://doi.org/10.1090/bproc/203","url":null,"abstract":"<p>We describe in terms of generators and relations the ring structure of the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R upper O left-parenthesis upper C 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:mi>O</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">RO(C_2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-graded <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C 2\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">C_2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivariant stable stems <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript star Superscript upper C 2\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mo>⋆<!-- ⋆ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">pi _star ^{C_2}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C 2\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">C_2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-equivariant sphere <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript star Superscript upper C 2 Baseline left-parenthesis double-struck upper S Subscript double-struck upper Q Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mo>⋆<!-- ⋆ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:msub>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">S</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"50 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139381659","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":"Existence of the solutions to the Brocard–Ramanujan problem for norm forms","authors":"Wataru Takeda","doi":"10.1090/bproc/181","DOIUrl":"https://doi.org/10.1090/bproc/181","url":null,"abstract":"<p>The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma script l right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(x,ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x squared minus 1 equals script l factorial\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>=</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">x^2-1=ell !</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma script l right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(x, ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, up to a unit factor, to the equations <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper K Baseline left-parenthesis x right-parenthesis equals script l factorial\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">N_K(x) = ell !</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N Subscript upper K\"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">N_K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are the norms of number fields <inline-formula content-type=\"math/mathml\">","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139269301","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":"A Banach space theoretical characterization of abelian 𝐶*-algebras","authors":"Ryotaro Tanaka","doi":"10.1090/bproc/175","DOIUrl":"https://doi.org/10.1090/bproc/175","url":null,"abstract":"<p>A Banach space theoretical characterization of abelian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>- algebras among all <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebras is given. As an application, it is shown that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\">\u0000 <mml:semantics>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">B</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebras (nonlinearly) isomorphic to each other with respect to the structure of Birkhoff-James orthogonality, and if either <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> or <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\">\u0000 <mml:semantics>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">B</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is abelian, then they are <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"asterisk\">\u0000 <mml:semantics>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 <mml:annotation encoding=\"application/x-tex\">*</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-isomorphic. Moreover, it is pointed out that the same kind of characterization is not valid for preduals of abelian von Neumann algebras.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121195333","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}