We prove that the ith graded pieces of the Orlik-Solomon algebras or Artinian Orlik-Terao algebras of resonance arrangements form a finitely generated FS^op-module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.
{"title":"Stability phenomena for resonance arrangements","authors":"Eric Ramos, N. Proudfoot","doi":"10.1090/bproc/71","DOIUrl":"https://doi.org/10.1090/bproc/71","url":null,"abstract":"We prove that the ith graded pieces of the Orlik-Solomon algebras or Artinian Orlik-Terao algebras of resonance arrangements form a finitely generated FS^op-module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"143 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123963055","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}
R. Anderson, Haosui Duanmu, David Schrittesser, W. Weiss
In [J. London Math. Soc. 69 (2004), pp. 258–272] Keisler and Sun leave open several questions regarding Loeb equivalence between internal probability spaces; specifically, whether under certain conditions, the Loeb measure construction applied to two such spaces gives rise to the same measure. We present answers to two of these questions, by giving two examples of probability spaces. Moreover, we reduce their third question to the following: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of their common Loeb extension? We also present a sufficient condition for a positive answer to this question.
{"title":"Loeb extension and Loeb equivalence","authors":"R. Anderson, Haosui Duanmu, David Schrittesser, W. Weiss","doi":"10.1090/bproc/78","DOIUrl":"https://doi.org/10.1090/bproc/78","url":null,"abstract":"In [J. London Math. Soc. 69 (2004), pp. 258–272] Keisler and Sun leave open several questions regarding Loeb equivalence between internal probability spaces; specifically, whether under certain conditions, the Loeb measure construction applied to two such spaces gives rise to the same measure. We present answers to two of these questions, by giving two examples of probability spaces. Moreover, we reduce their third question to the following: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of their common Loeb extension? We also present a sufficient condition for a positive answer to this question.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115330839","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}
We consider a two-sided Pompeiu type problem for a discrete group GG. We give necessary and sufficient conditions for a finite subset KK of GG to have the F(G)mathcal {F}(G)-Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for GG to be an ℓ2(G)ell ^2(G)-Pompeiu group.
{"title":"The two-sided Pompeiu problem for discrete groups","authors":"P. Linnell, M. Puls","doi":"10.1090/bproc/124","DOIUrl":"https://doi.org/10.1090/bproc/124","url":null,"abstract":"<p>We consider a two-sided Pompeiu type problem for a discrete group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We give necessary and sufficient conditions for a finite subset <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to have the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {F}(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Pompeiu property. Using group von Neumann algebra techniques, we give necessary and sufficient conditions for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to be an <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l squared left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ell ^2(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Pompeiu group.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127641156","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}
The differential Brauer monoid of a differential commutative ring R R is defined. Its elements are the isomorphism classes of differential Azumaya R R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of R R and R D R^D and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings.
定义了微分交换环R R的微分Brauer单群。它的元素是微分Azumaya R R代数的同构类，从张量积进行运算，前提是两个这样的代数是等价的，如果它们上面的矩阵代数具有入口微分，则它们是差分同构的。精细的鲍尔单线，是一个群，没有微分要求是一样的。然后由rr和rdr ^D的精细Brauer单群和Brauer单群的子单群确定微分Brauer单群，其基础Azumaya代数为矩阵环。
{"title":"Differential Brauer monoids","authors":"A. Magid","doi":"10.1090/bproc/162","DOIUrl":"https://doi.org/10.1090/bproc/162","url":null,"abstract":"The differential Brauer monoid of a differential commutative ring \u0000\u0000 \u0000 R\u0000 R\u0000 \u0000\u0000 is defined. Its elements are the isomorphism classes of differential Azumaya \u0000\u0000 \u0000 R\u0000 R\u0000 \u0000\u0000 algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of \u0000\u0000 \u0000 R\u0000 R\u0000 \u0000\u0000 and \u0000\u0000 \u0000 \u0000 R\u0000 D\u0000 \u0000 R^D\u0000 \u0000\u0000 and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114506818","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}
In this paper we consider interpolation in model spaces, $H^2 ominus B H^2$ with $B$ a Blaschke product. We study unions of interpolating sequences for two sequences that are far from each other in the pseudohyperbolic metric as well as two sequences that are close to each other in the pseudohyperbolic metric. The paper concludes with a discussion of the behavior of Frostman sequences under perturbations.
本文考虑模型空间中H^2 - H^2与B$ a Blaschke积的插值问题。研究了伪双曲度规中距离较远的两个序列和伪双曲度规中距离较近的两个序列的插值序列并。最后讨论了扰动作用下Frostman序列的行为。
{"title":"Interpolation in model spaces","authors":"P. Gorkin, B. Wick","doi":"10.1090/bproc/59","DOIUrl":"https://doi.org/10.1090/bproc/59","url":null,"abstract":"In this paper we consider interpolation in model spaces, $H^2 ominus B H^2$ with $B$ a Blaschke product. We study unions of interpolating sequences for two sequences that are far from each other in the pseudohyperbolic metric as well as two sequences that are close to each other in the pseudohyperbolic metric. The paper concludes with a discussion of the behavior of Frostman sequences under perturbations.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122723323","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}
Thomas Britz, A. Carey, F. Gesztesy, Roger Nichols, F. Sukochev, D. Zanin
For trace class operators $A, B in mathcal{B}_1(mathcal{H})$ ($mathcal{H}$ a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form [ {det}_{mathcal{H}} ((I_{mathcal{H}} - A) (I_{mathcal{H}} - B)) = {det}_{mathcal{H}} (I_{mathcal{H}} - A) {det}_{mathcal{H}} (I_{mathcal{H}} - B). ] When trace class operators are replaced by Hilbert--Schmidt operators $A, B in mathcal{B}_2(mathcal{H})$ and the Fredholm determinant ${det}_{mathcal{H}}(I_{mathcal{H}} - A)$, $A in mathcal{B}_1(mathcal{H})$, by the 2nd regularized Fredholm determinant ${det}_{mathcal{H},2}(I_{mathcal{H}} - A) = {det}_{mathcal{H}} ((I_{mathcal{H}} - A) exp(A))$, $A in mathcal{B}_2(mathcal{H})$, the product formula must be replaced by [ {det}_{mathcal{H},2} ((I_{mathcal{H}} - A) (I_{mathcal{H}} - B)) = {det}_{mathcal{H},2} (I_{mathcal{H}} - A) {det}_{mathcal{H},2} (I_{mathcal{H}} - B) exp(- {rm tr}(AB)). ] The product formula for the case of higher regularized Fredholm determinants ${det}_{mathcal{H},k}(I_{mathcal{H}} - A)$, $A in mathcal{B}_k(mathcal{H})$, $k in mathbb{N}$, $k geq 2$, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.
对于跟踪类算子$A, B in mathcal{B}_1(mathcal{H})$ ($mathcal{H}$一个复的，可分离的希尔伯特空间)，Fredholm行列式的乘积公式保持在熟悉的形式[ {det}_{mathcal{H}} ((I_{mathcal{H}} - A) (I_{mathcal{H}} - B)) = {det}_{mathcal{H}} (I_{mathcal{H}} - A) {det}_{mathcal{H}} (I_{mathcal{H}} - B). ]当跟踪类算子被Hilbert- Schmidt算子$A, B in mathcal{B}_2(mathcal{H})$和Fredholm行列式${det}_{mathcal{H}}(I_{mathcal{H}} - A)$, $A in mathcal{B}_1(mathcal{H})$取代时，由第2正则化Fredholm行列式${det}_{mathcal{H},2}(I_{mathcal{H}} - A) = {det}_{mathcal{H}} ((I_{mathcal{H}} - A) exp(A))$, $A in mathcal{B}_2(mathcal{H})$，乘积公式必须用[ {det}_{mathcal{H},2} ((I_{mathcal{H}} - A) (I_{mathcal{H}} - B)) = {det}_{mathcal{H},2} (I_{mathcal{H}} - A) {det}_{mathcal{H},2} (I_{mathcal{H}} - B) exp(- {rm tr}(AB)). ]代替更高正则化Fredholm行列式${det}_{mathcal{H},k}(I_{mathcal{H}} - A)$, $A in mathcal{B}_k(mathcal{H})$, $k in mathbb{N}$, $k geq 2$的乘积公式似乎不容易获得，因此本说明旨在填补文献中的这一空白。
{"title":"The product formula for regularized Fredholm determinants","authors":"Thomas Britz, A. Carey, F. Gesztesy, Roger Nichols, F. Sukochev, D. Zanin","doi":"10.1090/BPROC/70","DOIUrl":"https://doi.org/10.1090/BPROC/70","url":null,"abstract":"For trace class operators $A, B in mathcal{B}_1(mathcal{H})$ ($mathcal{H}$ a complex, separable Hilbert space), the product formula for Fredholm determinants holds in the familiar form [ {det}_{mathcal{H}} ((I_{mathcal{H}} - A) (I_{mathcal{H}} - B)) = {det}_{mathcal{H}} (I_{mathcal{H}} - A) {det}_{mathcal{H}} (I_{mathcal{H}} - B). ] When trace class operators are replaced by Hilbert--Schmidt operators $A, B in mathcal{B}_2(mathcal{H})$ and the Fredholm determinant ${det}_{mathcal{H}}(I_{mathcal{H}} - A)$, $A in mathcal{B}_1(mathcal{H})$, by the 2nd regularized Fredholm determinant ${det}_{mathcal{H},2}(I_{mathcal{H}} - A) = {det}_{mathcal{H}} ((I_{mathcal{H}} - A) exp(A))$, $A in mathcal{B}_2(mathcal{H})$, the product formula must be replaced by [ {det}_{mathcal{H},2} ((I_{mathcal{H}} - A) (I_{mathcal{H}} - B)) = {det}_{mathcal{H},2} (I_{mathcal{H}} - A) {det}_{mathcal{H},2} (I_{mathcal{H}} - B) exp(- {rm tr}(AB)). ] The product formula for the case of higher regularized Fredholm determinants ${det}_{mathcal{H},k}(I_{mathcal{H}} - A)$, $A in mathcal{B}_k(mathcal{H})$, $k in mathbb{N}$, $k geq 2$, does not seem to be easily accessible and hence this note aims at filling this gap in the literature.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116765260","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}
Let G G be a finite nonabelian group, and let ψ : G → G psi :Gto G be a homomorphism with abelian image. We show how ψ psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.
{"title":"Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures","authors":"Alan Koch","doi":"10.1090/BPROC/87","DOIUrl":"https://doi.org/10.1090/BPROC/87","url":null,"abstract":"Let \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000 be a finite nonabelian group, and let \u0000\u0000 \u0000 \u0000 ψ\u0000 :\u0000 G\u0000 →\u0000 G\u0000 \u0000 psi :Gto G\u0000 \u0000\u0000 be a homomorphism with abelian image. We show how \u0000\u0000 \u0000 ψ\u0000 psi\u0000 \u0000\u0000 gives rise to two Hopf-Galois structures on a Galois extension \u0000\u0000 \u0000 \u0000 L\u0000 \u0000 /\u0000 \u0000 K\u0000 \u0000 L/K\u0000 \u0000\u0000 with Galois group (isomorphic to) \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134547286","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}
Let kk be a field of characteristic p>0p > 0. For GG an elementary abelian pp-group, there exist collections of permutation modules such that if C∗C^* is any exact bounded complex whose terms are sums of copies of modules from the collection, then C∗C^* is contractible. A consequence is that if GG is any finite group whose Sylow pp-subgroups are not cyclic or quaternion, and if
设k k为特征p > 0 p > 0的域。对于G G一个初等阿贝尔p -群，存在这样的置换模集合，如果C * C^*是任何精确有界复，其项是该集合中模的副本的和，则C * C^*是可缩并的。一个结果是，如果G G是任何有限群，其Sylow p p -子群不是循环或四元数，并且如果C * C^*是一个有界的精确复，使得每个C * C^i是一维模与投影模的直接和，则C * C^*是可缩并的。
{"title":"Bounded complexes of permutation modules","authors":"D. Benson, J. Carlson","doi":"10.1090/bproc/102","DOIUrl":"https://doi.org/10.1090/bproc/102","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a field of characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p > 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. For <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> an elementary abelian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-group, there exist collections of permutation modules such that if <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> is any exact bounded complex whose terms are sums of copies of modules from the collection, then <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> is contractible. A consequence is that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is any finite group whose Sylow <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-subgroups are not cyclic or quaternion, and if <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:","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123748095","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}
Let EE be a countable directed graph that is amplified in the sense that whenever there is an edge from vv to ww, there are infinitely many edges from vv to ww. We show that EE can be recovered from C∗(E)C^*(E) together with its canonical gauge-action, and also from LK(E)
{"title":"Amplified graph C*-algebras II: Reconstruction","authors":"S. Eilers, Efren Ruiz, A. Sims","doi":"10.1090/bproc/112","DOIUrl":"https://doi.org/10.1090/bproc/112","url":null,"abstract":"<p>Let <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> be a countable directed graph that is amplified in the sense that whenever there is an edge from <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\u0000 <mml:semantics>\u0000 <mml:mi>v</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"w\">\u0000 <mml:semantics>\u0000 <mml:mi>w</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">w</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, there are infinitely many edges from <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\u0000 <mml:semantics>\u0000 <mml:mi>v</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"w\">\u0000 <mml:semantics>\u0000 <mml:mi>w</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">w</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We show that <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> can be recovered from <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk Baseline left-parenthesis upper E right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">C^*(E)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> together with its canonical gauge-action, and also from <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript double-struck upper K Baseline left-parenthesis upper E right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">K</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>E</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132616950","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}
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