Pub Date : 2025-03-01DOI: 10.1016/j.exmath.2024.125586
M.-K. Chuah , C.A. Cremonini , R. Fioresi
We realize the irreducible representations of a compact Lie supergroup , with a contragredient simple Lie superalgebra, in the space of square integrable (in the sense of Berezin) holomorphic sections on , is the real torus in the complexification of . We give an explicit realization of unitary representations when .
{"title":"Harmonic analysis of compact Lie supergroups","authors":"M.-K. Chuah , C.A. Cremonini , R. Fioresi","doi":"10.1016/j.exmath.2024.125586","DOIUrl":"10.1016/j.exmath.2024.125586","url":null,"abstract":"<div><div>We realize the irreducible representations of a compact Lie supergroup <span><math><mi>G</mi></math></span>, with a contragredient simple Lie superalgebra, in the space of square integrable (in the sense of Berezin) holomorphic sections on <span><math><mrow><mi>X</mi><mo>=</mo><mi>G</mi><mi>A</mi></mrow></math></span>, <span><math><mi>A</mi></math></span> is the real torus in the complexification of <span><math><mi>G</mi></math></span>. We give an explicit realization of unitary representations when <span><math><mrow><mi>G</mi><mo>=</mo><mi>SU</mi><mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125586"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.exmath.2024.125572
Glenn D. Appleby
Thirty years ago, I had just completed my Ph.D. under Varadarajan when, as part of a subsequent reading course, he and I considered a generalization of von Neumann entropy, here called a matrix entropy, computed by using the classical entropy function on the diagonals of density matrices. I had asked whether the value of von Neumann entropy was the maximum of the matrix entropy on a given unitary equivalence class. Varadarajan soon sketched a proof of this, which is presented here. It serves as a nice way to see classical entropy sitting in the von Neumann entropy context, and a reminder for this short note’s author of a pleasant time spent working with a remarkable scholar and teacher.
{"title":"A variation on von Neumann Entropy and a result of Varadarajan","authors":"Glenn D. Appleby","doi":"10.1016/j.exmath.2024.125572","DOIUrl":"10.1016/j.exmath.2024.125572","url":null,"abstract":"<div><div>Thirty years ago, I had just completed my Ph.D. under Varadarajan when, as part of a subsequent reading course, he and I considered a generalization of von Neumann entropy, here called a <em>matrix entropy</em>, computed by using the classical entropy function on the diagonals of density matrices. I had asked whether the value of von Neumann entropy was the maximum of the matrix entropy on a given unitary equivalence class. Varadarajan soon sketched a proof of this, which is presented here. It serves as a nice way to see classical entropy sitting in the von Neumann entropy context, and a reminder for this short note’s author of a pleasant time spent working with a remarkable scholar and teacher.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125572"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140796018","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}
Pub Date : 2025-03-01DOI: 10.1016/j.exmath.2024.125573
Trond Digernes
We review some topics from our collaboration with V.S. Varadarajan.
我们回顾了我们与V.S. Varadarajan合作的一些主题。
{"title":"A short review of finite approximations and unconventional physics","authors":"Trond Digernes","doi":"10.1016/j.exmath.2024.125573","DOIUrl":"10.1016/j.exmath.2024.125573","url":null,"abstract":"<div><div>We review some topics from our collaboration with V.S. Varadarajan.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125573"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141052781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.exmath.2025.125660
Maria Stella Adamo , Karl-Hermann Neeb , Jonas Schober
We present a novel perspective on reflection positivity on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on for a suitable involution. For the strip, reflection positivity naturally connects with Kubo–Martin–Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.
{"title":"Reflection positivity and its relation to disc, half plane and the strip","authors":"Maria Stella Adamo , Karl-Hermann Neeb , Jonas Schober","doi":"10.1016/j.exmath.2025.125660","DOIUrl":"10.1016/j.exmath.2025.125660","url":null,"abstract":"<div><div>We present a novel perspective on reflection positivity on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> for a suitable involution. For the strip, reflection positivity naturally connects with Kubo–Martin–Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125660"},"PeriodicalIF":0.8,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143579109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-04DOI: 10.1016/j.exmath.2025.125659
Luc Guyot
Let be the ring of univariate polynomials over and denote by its stable rank in the sense of Bass. Grunewald, Mennicke and Vaserstein proved that As the inequality follows immediately from Bass’s stable range theorem, the above identity is equivalent to the existence of a non-stable unimodular row of size 3. This note addresses minor errors found in the existing proof of the latter fact. Using the same methods, we show that the unimodular row is not stable.
{"title":"The stable rank of Z[x] is 3","authors":"Luc Guyot","doi":"10.1016/j.exmath.2025.125659","DOIUrl":"10.1016/j.exmath.2025.125659","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></mrow></math></span> be the ring of univariate polynomials over <span><math><mi>Z</mi></math></span> and denote by <span><math><mrow><mo>sr</mo><mrow><mo>(</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>)</mo></mrow></mrow></math></span> its stable rank in the sense of Bass. Grunewald, Mennicke and Vaserstein proved that <span><math><mrow><mo>sr</mo><mrow><mo>(</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mo>.</mo></mrow></math></span> As the inequality <span><math><mrow><mo>sr</mo><mrow><mo>(</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span> follows immediately from Bass’s stable range theorem, the above identity is equivalent to the existence of a non-stable unimodular row of size 3. This note addresses minor errors found in the existing proof of the latter fact. Using the same methods, we show that the unimodular row <span><math><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>16</mn><mo>)</mo></mrow></math></span> is not stable.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125659"},"PeriodicalIF":0.8,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349097","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}
Pub Date : 2025-02-01DOI: 10.1016/j.exmath.2024.125635
Mykola Moroz
We consider the representation of real numbers by alternating Perron series (-representation), which is a generalization of representations of real numbers by Ostrogradsky–Sierpiński–Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating Lüroth series, etc. Namely, we prove the basic topological and metric properties of -representation and find the relationship between -representation and -representation in some measure theory problems.
{"title":"Representations of real numbers by alternating Perron series and their geometry","authors":"Mykola Moroz","doi":"10.1016/j.exmath.2024.125635","DOIUrl":"10.1016/j.exmath.2024.125635","url":null,"abstract":"<div><div>We consider the representation of real numbers by alternating Perron series (<span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation), which is a generalization of representations of real numbers by Ostrogradsky–Sierpiński–Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating Lüroth series, etc. Namely, we prove the basic topological and metric properties of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation and find the relationship between <span><math><mi>P</mi></math></span>-representation and <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation in some measure theory problems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125635"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133974","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}
Pub Date : 2025-02-01DOI: 10.1016/j.exmath.2024.125634
Kai Rajala
Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In “Extremal length and functional completion”, Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings:
(1) Extremal length and uniformization of metric surfaces.
(2) Extremal length of families of surfaces and quasiconformal maps between -dimensional spaces.
(3) Schramm’s transboundary extremal length and conformal maps between multiply connected plane domains.
{"title":"Extremal length and duality","authors":"Kai Rajala","doi":"10.1016/j.exmath.2024.125634","DOIUrl":"10.1016/j.exmath.2024.125634","url":null,"abstract":"<div><div>Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In “Extremal length and functional completion”, Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings:</div><div>(1) Extremal length and uniformization of metric surfaces.</div><div>(2) Extremal length of families of surfaces and quasiconformal maps between <span><math><mi>n</mi></math></span>-dimensional spaces.</div><div>(3) Schramm’s transboundary extremal length and conformal maps between multiply connected plane domains.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125634"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.exmath.2024.125633
Mohith Raju Nagaraju
Roughly, a conformal tiling of a Riemann surface is a tiling where each tile is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers and Stephenson constructed an edge-to-edge conformal tiling of the complex plane using conformally regular pentagons. In contrast, we show that for all , there is no edge-to-edge conformal tiling of the complex plane using conformally regular -gons. More generally, we discuss a relationship between the combinatorial curvature at each vertex of the conformal tiling and the universal cover (sphere, plane, or disc) of the underlying Riemann surface. This result follows from the work of Stone (1976) and Oh (2005) through a rich interplay between Riemannian geometry and combinatorial geometry. We provide an exposition of these proofs and some new applications to conformal tilings.
{"title":"Conformal tilings, combinatorial curvature, and the type problem","authors":"Mohith Raju Nagaraju","doi":"10.1016/j.exmath.2024.125633","DOIUrl":"10.1016/j.exmath.2024.125633","url":null,"abstract":"<div><div>Roughly, a conformal tiling of a Riemann surface is a tiling where each tile is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers and Stephenson constructed an edge-to-edge conformal tiling of the complex plane using conformally regular pentagons. In contrast, we show that for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow></math></span>, there is no edge-to-edge conformal tiling of the complex plane using conformally regular <span><math><mi>n</mi></math></span>-gons. More generally, we discuss a relationship between the combinatorial curvature at each vertex of the conformal tiling and the universal cover (sphere, plane, or disc) of the underlying Riemann surface. This result follows from the work of Stone (1976) and Oh (2005) through a rich interplay between Riemannian geometry and combinatorial geometry. We provide an exposition of these proofs and some new applications to conformal tilings.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125633"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133973","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}
Pub Date : 2025-01-30DOI: 10.1016/j.exmath.2025.125658
Tsiu-Kwen Lee , Jheng-Huei Lin
Motivated by some recent results on Lie ideals, it is proved that if is a Lie ideal of a simple ring with center , then , for some noncentral , or , which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let be a prime ring with extended centroid . We completely characterize Lie ideals and elements of such that contains a nonzero ideal of . Given noncentral Lie ideals of , it is proved that if and only if for any noncentral element . As a consequence, we characterize noncentral Lie ideals with such that contains a nonzero ideal of . Finally, we characterize noncentral Lie ideals ’s and ’s satisfying
{"title":"Commutators and products of Lie ideals of prime rings","authors":"Tsiu-Kwen Lee , Jheng-Huei Lin","doi":"10.1016/j.exmath.2025.125658","DOIUrl":"10.1016/j.exmath.2025.125658","url":null,"abstract":"<div><div>Motivated by some recent results on Lie ideals, it is proved that if <span><math><mi>L</mi></math></span> is a Lie ideal of a simple ring <span><math><mi>R</mi></math></span> with center <span><math><mrow><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, then <span><math><mrow><mi>L</mi><mo>⊆</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>L</mi><mo>=</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mi>a</mi><mo>+</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for some noncentral <span><math><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow></math></span>, or <span><math><mrow><mrow><mo>[</mo><mi>R</mi><mo>,</mo><mi>R</mi><mo>]</mo></mrow><mo>⊆</mo><mi>L</mi></mrow></math></span>, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let <span><math><mi>R</mi></math></span> be a prime ring with extended centroid <span><math><mi>C</mi></math></span>. We completely characterize Lie ideals <span><math><mi>L</mi></math></span> and elements <span><math><mi>a</mi></math></span> of <span><math><mi>R</mi></math></span> such that <span><math><mrow><mi>L</mi><mo>+</mo><mi>a</mi><mi>L</mi></mrow></math></span> contains a nonzero ideal of <span><math><mi>R</mi></math></span>. Given noncentral Lie ideals <span><math><mrow><mi>K</mi><mo>,</mo><mi>L</mi></mrow></math></span> of <span><math><mi>R</mi></math></span>, it is proved that <span><math><mrow><mrow><mo>[</mo><mi>K</mi><mo>,</mo><mi>L</mi><mo>]</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> if and only if <span><math><mrow><mi>K</mi><mi>C</mi><mo>=</mo><mi>L</mi><mi>C</mi><mo>=</mo><mi>C</mi><mi>a</mi><mo>+</mo><mi>C</mi></mrow></math></span> for any noncentral element <span><math><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow></math></span>. As a consequence, we characterize noncentral Lie ideals <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> with <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> contains a nonzero ideal of <span><math><mi>R</mi></math></span>. Finally, we characterize noncentral Lie ideals <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>’s and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>’s satisfying <span><math><mrow><mrow><mo>[</mo><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>K</mi></mr","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125658"},"PeriodicalIF":0.8,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349059","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}
Pub Date : 2025-01-25DOI: 10.1016/j.exmath.2025.125657
Konrad Schmüdgen
Suppose that is a finitely generated commutative unital real algebra and is a closed subset of the set of characters of . We study the following problem: When is each linear functional an integral with respect to some signed Radon measure on supported by the set ? A complete characterization of these sets and algebras by necessary and sufficient conditions is given. The result is applied to the polynomial algebra and subsets of .
{"title":"On moment functionals with signed representing measures","authors":"Konrad Schmüdgen","doi":"10.1016/j.exmath.2025.125657","DOIUrl":"10.1016/j.exmath.2025.125657","url":null,"abstract":"<div><div>Suppose that <span><math><mi>A</mi></math></span> is a finitely generated commutative unital real algebra and <span><math><mi>K</mi></math></span> is a closed subset of the set <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> of characters of <span><math><mi>A</mi></math></span>. We study the following problem: When is <em>each</em> linear functional <span><math><mrow><mi>L</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>R</mi></mrow></math></span> an integral with respect to some signed Radon measure on <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> supported by the set <span><math><mi>K</mi></math></span>? A complete characterization of these sets <span><math><mi>K</mi></math></span> and algebras <span><math><mi>A</mi></math></span> by necessary and sufficient conditions is given. The result is applied to the polynomial algebra <span><math><mrow><mi>R</mi><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> and subsets <span><math><mi>K</mi></math></span> of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125657"},"PeriodicalIF":0.8,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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