Von Neumann spectra near the spectral gap
Identifieur interne : 002751 ( Istex/Corpus ); précédent : 002750; suivant : 002752Von Neumann spectra near the spectral gap
Auteurs : Alan L. Carey ; Thierry Coulhon ; Varghese Mathai ; John PhillipsSource :
- Bulletin des sciences mathematiques [ 0007-4497 ] ; 1998.
English descriptors
- KwdEn :
- 58G11, 58G18 and 58G25, Acyclic riemannian manifold, Analytic laplacian, Analytic torsion, Betti numbers, Carey, Casimir operator, Combinatorial, Combinatorial counterpart, Combinatorial laplacian, Compact subsets, Convergence, Converges, Determinant, Determinant class, Differential forms, Dimr, Fundamental group, Heat kernel, Heat kernels, Hyperbolic manifold, Hyperbolic manifolds, Inequality, Integral kernel, Laplacian, Large time asymptotics, Lemma, Mathai, Meromorphic continuation, Mesh, Metric, Neumann, Neumann algebra, Neumann determinant, Neumann determinants, Neumann spectra, Neumann trace, Novikov-Shubin invariants, Partial zeta functions, Plancherel measure, Positive decay, Previous theorem, Rham, Riemannian, Riemannian manifold, Right hand side, Sciences mathbmatiques, Spectral density function, Subset, Theta, Theta functions, Tome, Torsion, Triangulation, Zeta, Zeta function, Zeta functions, bottom of the spectrum, torsion, von Neumann determinants.
- Teeft :
- Acyclic riemannian manifold, Analytic laplacian, Analytic torsion, Betti numbers, Carey, Casimir operator, Combinatorial, Combinatorial counterpart, Combinatorial laplacian, Compact subsets, Convergence, Converges, Determinant, Determinant class, Differential forms, Dimr, Fundamental group, Heat kernel, Hyperbolic manifold, Hyperbolic manifolds, Inequality, Integral kernel, Laplacian, Large time asymptotics, Lemma, Mathai, Meromorphic continuation, Mesh, Metric, Neumann, Neumann algebra, Neumann determinant, Neumann determinants, Neumann spectra, Neumann trace, Partial zeta functions, Plancherel measure, Positive decay, Previous theorem, Rham, Riemannian, Riemannian manifold, Right hand side, Sciences mathbmatiques, Spectral density function, Subset, Theta, Theta functions, Tome, Torsion, Triangulation, Zeta, Zeta function, Zeta functions.
Abstract
Abstract: In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.
Url:
DOI: 10.1016/S0007-4497(98)80088-2
Links to Exploration step
ISTEX:D3A19ABC400F673991C245370E203D5A751A342BLe document en format XML
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Carey</name><affiliation><mods:affiliation>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</mods:affiliation></affiliation></author><author><name sortKey="Coulhon, Thierry" sort="Coulhon, Thierry" uniqKey="Coulhon T" first="Thierry" last="Coulhon">Thierry Coulhon</name><affiliation><mods:affiliation>Département de Mathématiques, Université de Cergy-Pontoise, 95033 Cergy-Pontoise cedex, France</mods:affiliation></affiliation></author><author><name sortKey="Mathai, Varghese" sort="Mathai, Varghese" uniqKey="Mathai V" first="Varghese" last="Mathai">Varghese Mathai</name><affiliation><mods:affiliation>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</mods:affiliation></affiliation></author><author><name sortKey="Phillips, John" sort="Phillips, John" uniqKey="Phillips J" first="John" last="Phillips">John Phillips</name><affiliation><mods:affiliation>Department of Mathematics, University of Victoria. 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Carey</name><affiliation><mods:affiliation>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</mods:affiliation></affiliation></author><author><name sortKey="Coulhon, Thierry" sort="Coulhon, Thierry" uniqKey="Coulhon T" first="Thierry" last="Coulhon">Thierry Coulhon</name><affiliation><mods:affiliation>Département de Mathématiques, Université de Cergy-Pontoise, 95033 Cergy-Pontoise cedex, France</mods:affiliation></affiliation></author><author><name sortKey="Mathai, Varghese" sort="Mathai, Varghese" uniqKey="Mathai V" first="Varghese" last="Mathai">Varghese Mathai</name><affiliation><mods:affiliation>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</mods:affiliation></affiliation></author><author><name sortKey="Phillips, John" sort="Phillips, John" uniqKey="Phillips J" first="John" last="Phillips">John Phillips</name><affiliation><mods:affiliation>Department of Mathematics, University of Victoria. 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These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</div></front></TEI><istex><corpusName>elsevier</corpusName><keywords><teeft><json:string>combinatorial</json:string><json:string>triangulation</json:string><json:string>zeta</json:string><json:string>laplacian</json:string><json:string>converges</json:string><json:string>neumann</json:string><json:string>tome</json:string><json:string>theta</json:string><json:string>zeta function</json:string><json:string>riemannian</json:string><json:string>neumann spectra</json:string><json:string>dimr</json:string><json:string>zeta functions</json:string><json:string>subset</json:string><json:string>determinant</json:string><json:string>positive decay</json:string><json:string>metric</json:string><json:string>theta functions</json:string><json:string>torsion</json:string><json:string>mathai</json:string><json:string>convergence</json:string><json:string>rham</json:string><json:string>neumann determinant</json:string><json:string>compact subsets</json:string><json:string>combinatorial counterpart</json:string><json:string>combinatorial laplacian</json:string><json:string>inequality</json:string><json:string>riemannian manifold</json:string><json:string>fundamental group</json:string><json:string>neumann determinants</json:string><json:string>sciences mathbmatiques</json:string><json:string>meromorphic continuation</json:string><json:string>analytic laplacian</json:string><json:string>lemma</json:string><json:string>casimir operator</json:string><json:string>spectral density function</json:string><json:string>large time asymptotics</json:string><json:string>betti numbers</json:string><json:string>neumann trace</json:string><json:string>heat kernel</json:string><json:string>right hand side</json:string><json:string>hyperbolic manifolds</json:string><json:string>hyperbolic manifold</json:string><json:string>neumann algebra</json:string><json:string>plancherel measure</json:string><json:string>previous theorem</json:string><json:string>integral kernel</json:string><json:string>acyclic riemannian manifold</json:string><json:string>determinant class</json:string><json:string>differential forms</json:string><json:string>partial zeta functions</json:string><json:string>analytic torsion</json:string><json:string>mesh</json:string><json:string>carey</json:string></teeft></keywords><author><json:item><name>Alan L. Carey</name><affiliations><json:string>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</json:string></affiliations></json:item><json:item><name>Thierry Coulhon</name><affiliations><json:string>Département de Mathématiques, Université de Cergy-Pontoise, 95033 Cergy-Pontoise cedex, France</json:string></affiliations></json:item><json:item><name>Varghese Mathai</name><affiliations><json:string>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</json:string></affiliations></json:item><json:item><name>John Phillips</name><affiliations><json:string>Department of Mathematics, University of Victoria. Victoria, BC, Canada</json:string><json:string>(∗)Manuscript presented by G. 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These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</abstract><qualityIndicators><score>8.452</score><pdfWordCount>9566</pdfWordCount><pdfCharCount>45979</pdfCharCount><pdfVersion>1.3</pdfVersion><pdfPageCount>40</pdfPageCount><pdfPageSize>432 x 648 pts</pdfPageSize><refBibsNative>true</refBibsNative><abstractWordCount>121</abstractWordCount><abstractCharCount>781</abstractCharCount><keywordCount>7</keywordCount></qualityIndicators><title>Von Neumann spectra near the spectral gap</title><pii><json:string>S0007-4497(98)80088-2</json:string></pii><genre><json:string>research-article</json:string></genre><serie><title>Cours de l'École d'été de probabilités de Saint-Flour, 1988</title><language><json:string>unknown</json:string></language><pages><first>4</first><last>112</last></pages></serie><host><title>Bulletin des sciences mathematiques</title><language><json:string>unknown</json:string></language><publicationDate>1998</publicationDate><issn><json:string>0007-4497</json:string></issn><pii><json:string>S0007-4497(00)X0001-2</json:string></pii><volume>122</volume><issue>3</issue><pages><first>203</first><last>242</last></pages><genre><json:string>journal</json:string></genre></host><namedEntities><unitex><date><json:string>1201</json:string><json:string>1998</json:string></date><geogName></geogName><orgName><json:string>Department of Mathematics, University of Victoria</json:string><json:string>Department of Pure Mathematics, University of Adelaide, ADELAIDE, South Australia, Australia</json:string></orgName><orgName_funder></orgName_funder><orgName_provider></orgName_provider><persName><json:string>A. 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Victoria, BC, Canada</affiliation><affiliation>(∗)Manuscript presented by G. Pisier, received in March 1996.</affiliation></author><idno type="istex">D3A19ABC400F673991C245370E203D5A751A342B</idno><idno type="DOI">10.1016/S0007-4497(98)80088-2</idno><idno type="PII">S0007-4497(98)80088-2</idno><idno type="ArticleID">98800882</idno></analytic><monogr><title level="j">Bulletin des sciences mathematiques</title><title level="j" type="abbrev">BULSCI</title><idno type="pISSN">0007-4497</idno><idno type="PII">S0007-4497(00)X0001-2</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="1998"></date><biblScope unit="volume">122</biblScope><biblScope unit="issue">3</biblScope><biblScope unit="page" from="203">203</biblScope><biblScope unit="page" to="242">242</biblScope></imprint></monogr></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1998</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</p></abstract><textClass xml:lang="en"><keywords scheme="keyword"><list><head>Keywords</head><item><term>Heat kernels</term></item><item><term>bottom of the spectrum</term></item><item><term>Novikov-Shubin invariants</term></item><item><term>von Neumann determinants</term></item><item><term>torsion</term></item></list></keywords></textClass><textClass xml:lang="en"><keywords scheme="keyword"><list><head>MSC</head><item><term>58G11</term></item><item><term>58G18 and 58G25</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1998">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/D3A19ABC400F673991C245370E203D5A751A342B/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Elsevier, elements deleted: tail"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//ES//DTD journal article DTD version 4.5.2//EN//XML" URI="art452.dtd" name="istex:docType"></istex:docType><istex:document><converted-article version="4.5.2" docsubtype="fla" xml:lang="en"><item-info><jid>BULSCI</jid><aid>98800882</aid><ce:pii>S0007-4497(98)80088-2</ce:pii><ce:doi>10.1016/S0007-4497(98)80088-2</ce:doi><ce:copyright type="unknown" year="1998"></ce:copyright></item-info><head><ce:title>Von Neumann spectra near the spectral gap</ce:title><ce:author-group><ce:author><ce:given-name>Alan L.</ce:given-name><ce:surname>Carey</ce:surname><ce:cross-ref refid="AFF1"><ce:sup loc="post">1</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>Thierry</ce:given-name><ce:surname>Coulhon</ce:surname><ce:cross-ref refid="AFF2"><ce:sup loc="post">2</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>Varghese</ce:given-name><ce:surname>Mathai</ce:surname><ce:cross-ref refid="AFF3"><ce:sup loc="post">3</ce:sup></ce:cross-ref></ce:author><ce:author><ce:given-name>John</ce:given-name><ce:surname>Phillips</ce:surname><ce:cross-ref refid="AFF4"><ce:sup loc="post">4</ce:sup></ce:cross-ref><ce:cross-ref refid="FN1"><ce:sup loc="post">(∗)</ce:sup></ce:cross-ref></ce:author><ce:affiliation id="AFF1"><ce:label>a</ce:label><ce:textfn>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</ce:textfn></ce:affiliation><ce:affiliation id="AFF2"><ce:label>b</ce:label><ce:textfn>Département de Mathématiques, Université de Cergy-Pontoise, 95033 Cergy-Pontoise cedex, France</ce:textfn></ce:affiliation><ce:affiliation id="AFF3"><ce:label>c</ce:label><ce:textfn>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</ce:textfn></ce:affiliation><ce:affiliation id="AFF4"><ce:label>d</ce:label><ce:textfn>Department of Mathematics, University of Victoria. Victoria, BC, Canada</ce:textfn></ce:affiliation><ce:footnote id="FN1"><ce:label>(∗)</ce:label><ce:note-para>Manuscript presented by <ce:small-caps>G. Pisier</ce:small-caps>, received in March 1996.</ce:note-para></ce:footnote></ce:author-group><ce:abstract class="author"><ce:section-title>Abstract</ce:section-title><ce:abstract-sec><ce:simple-para view="all" id="simple-para.0010">In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on <ce:italic>L</ce:italic><ce:sup loc="post">2</ce:sup> differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain <ce:italic>L</ce:italic><ce:sup loc="post">2</ce:sup> theta and <ce:italic>L</ce:italic><ce:sup loc="post">2</ce:sup> zera functions defined by metric dependent combinatorial Laplacians acting on <ce:italic>L</ce:italic><ce:sup loc="post">2</ce:sup> cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</ce:simple-para></ce:abstract-sec></ce:abstract><ce:keywords class="keyword"><ce:section-title>Keywords</ce:section-title><ce:keyword><ce:text>Heat kernels</ce:text></ce:keyword><ce:keyword><ce:text>bottom of the spectrum</ce:text></ce:keyword><ce:keyword><ce:text>Novikov-Shubin invariants</ce:text></ce:keyword><ce:keyword><ce:text>von Neumann determinants</ce:text></ce:keyword><ce:keyword><ce:text>torsion</ce:text></ce:keyword></ce:keywords><ce:keywords class="msc"><ce:section-title>MSC</ce:section-title><ce:keyword><ce:text>58G11</ce:text></ce:keyword><ce:keyword><ce:text>58G18 and 58G25</ce:text></ce:keyword></ce:keywords></head></converted-article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Von Neumann spectra near the spectral gap</title></titleInfo><titleInfo type="alternative" lang="en" contentType="CDATA"><title>Von Neumann spectra near the spectral gap</title></titleInfo><name type="personal"><namePart type="given">Alan L.</namePart><namePart type="family">Carey</namePart><affiliation>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Thierry</namePart><namePart type="family">Coulhon</namePart><affiliation>Département de Mathématiques, Université de Cergy-Pontoise, 95033 Cergy-Pontoise cedex, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Varghese</namePart><namePart type="family">Mathai</namePart><affiliation>Department of Pure Mathematics, University of Adelaide, Adelaide, South Australia, Australia</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">John</namePart><namePart type="family">Phillips</namePart><affiliation>Department of Mathematics, University of Victoria. Victoria, BC, Canada</affiliation><affiliation>(∗)Manuscript presented by G. Pisier, received in March 1996.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="Full-length article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">1998</dateIssued><copyrightDate encoding="w3cdtf">1998</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><abstract lang="en">Abstract: In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy invariants in this case. In the presence of a spectral gap, they differ in character and value from the Novikov-Shubin invariants. Under a positivity assumption on these invariants, we prove that certain L2 theta and L2 zera functions defined by metric dependent combinatorial Laplacians acting on L2 cochains associated with a triangulation of the manifold, converge uniformly to their analytic counterparts, as the mesh of the triangulation goes to zero.</abstract><subject lang="en"><genre>Keywords</genre><topic>Heat kernels</topic><topic>bottom of the spectrum</topic><topic>Novikov-Shubin invariants</topic><topic>von Neumann determinants</topic><topic>torsion</topic></subject><subject lang="en"><genre>MSC</genre><topic>58G11</topic><topic>58G18 and 58G25</topic></subject><relatedItem type="host"><titleInfo><title>Bulletin des sciences mathematiques</title></titleInfo><titleInfo type="abbreviated"><title>BULSCI</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>ELSEVIER</publisher><dateIssued encoding="w3cdtf">199804</dateIssued></originInfo><identifier type="ISSN">0007-4497</identifier><identifier type="PII">S0007-4497(00)X0001-2</identifier><part><date>199804</date><detail type="volume"><number>122</number><caption>vol.</caption></detail><detail type="issue"><number>3</number><caption>no.</caption></detail><extent unit="issue-pages"><start>163</start><end>250</end></extent><extent unit="pages"><start>203</start><end>242</end></extent></part></relatedItem><identifier type="istex">D3A19ABC400F673991C245370E203D5A751A342B</identifier><identifier type="ark">ark:/67375/6H6-LP9GCR9L-X</identifier><identifier type="DOI">10.1016/S0007-4497(98)80088-2</identifier><identifier type="PII">S0007-4497(98)80088-2</identifier><identifier type="ArticleID">98800882</identifier><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-HKKZVM7B-M">elsevier</recordContentSource></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/D3A19ABC400F673991C245370E203D5A751A342B/metadata/json</uri></json:item></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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