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The uniqueness-of-norm problem for Calkin algebras Calkin代数的范数唯一性问题
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2015-07-29 DOI: 10.3318/PRIA.2015.115.14
Richard Skillicorn
{"title":"The uniqueness-of-norm problem for Calkin algebras","authors":"Richard Skillicorn","doi":"10.3318/PRIA.2015.115.14","DOIUrl":"https://doi.org/10.3318/PRIA.2015.115.14","url":null,"abstract":"We examine the question of whether the Calkin algebra of a Banach space must have a unique complete algebra norm. We present a survey of known results, and make the observation that a recent Banach space construction of Argyros and Motakis (preprint, 2015) provides the first negative answer. The parallel question for the weak Calkin algebra also has a negative answer; we demonstrate this using a Banach space of Read (J. London Math. Soc. 1989).","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"193 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121738730","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}
引用次数: 2
Ball generated property of direct sums of Banach spaces Banach空间直接和的球生成性质
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2015-02-22 DOI: 10.3318/PRIA.2015.115.13
Jan-David Hardtke
{"title":"Ball generated property of direct sums of Banach spaces","authors":"Jan-David Hardtke","doi":"10.3318/PRIA.2015.115.13","DOIUrl":"https://doi.org/10.3318/PRIA.2015.115.13","url":null,"abstract":"A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $ell^p$-sums for $1<p<infty$. We will show here that for any absolute, normalised norm $|cdot|_E$ on $mathbb{R}^2$ satisfying a certain smoothness condition the direct sum $Xoplus_E Y$ of two Banach spaces $X$ and $Y$ with respect to $|cdot|_E$ enjoys the BGP whenever $X$ and $Y$ have the BGP.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131685384","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}
引用次数: 1
Operators Approximable by Hypercyclic Operators 由超循环算子逼近的算子
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2014-10-27 DOI: 10.3318/PRIA.2015.115.08
J. Boland
{"title":"Operators Approximable by Hypercyclic Operators","authors":"J. Boland","doi":"10.3318/PRIA.2015.115.08","DOIUrl":"https://doi.org/10.3318/PRIA.2015.115.08","url":null,"abstract":"We show that operators on a separable infinite dimensional Banach space $X$ of the form $I +S$, where $S$ is an operator with dense generalised kernel, must lie in the norm closure of the hypercyclic operators on $X$, in fact in the closure of the mixing operators.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"198 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114977661","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}
引用次数: 0
The Haagerup Property for Drinfeld Doubles Drinfeld double的Haagerup性质
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2014-09-04 DOI: 10.3318/PRIA.2015.115.09
Sutanu Roy
{"title":"The Haagerup Property for Drinfeld Doubles","authors":"Sutanu Roy","doi":"10.3318/PRIA.2015.115.09","DOIUrl":"https://doi.org/10.3318/PRIA.2015.115.09","url":null,"abstract":"We show that Drinfeld's double group construction for locally compact quantum groups preserves the Haagerup property. This shows that the Drinfeld doubles of the quantum groups, $C_{0}(mathbb{F}_{2})$, $SU_{q}(2)$, $SU_{q}(1,1)_{text{ext}}$, quantum $ax+b$, quantum $az+b$, and $E_{q}(2)$ have the Haagerup property.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126786395","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}
引用次数: 1
Isolated spectral points and Koliha-Drazin invertible elements in quotient Banach algebras and homomorphism ranges 商Banach代数中的孤立谱点和Koliha-Drazin可逆元及同态范围
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2014-03-14 DOI: 10.3318/PRIA.2015.115.12
Enrico Boasso
{"title":"Isolated spectral points and Koliha-Drazin invertible elements in quotient Banach algebras and homomorphism ranges","authors":"Enrico Boasso","doi":"10.3318/PRIA.2015.115.12","DOIUrl":"https://doi.org/10.3318/PRIA.2015.115.12","url":null,"abstract":"In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133264066","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}
引用次数: 25
Vector Bundles on the Projective Line and Finite Domination of Chain Complexes 投影线上的向量束与链式配合物的有限支配
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2014-03-10 DOI: 10.3318/pria.2015.115.2
T. Huettemann
{"title":"Vector Bundles on the Projective Line and Finite Domination of Chain Complexes","authors":"T. Huettemann","doi":"10.3318/pria.2015.115.2","DOIUrl":"https://doi.org/10.3318/pria.2015.115.2","url":null,"abstract":"Finitely dominated chain complexes over a Laurent polynomial ring in one indeterminate are characterised by vanishing of their Novikov homology. We present an algebro-geometric approach to this result, based on extension of chain complexes to sheaves on the projective line. We also discuss the K-theoretical obstruction to extension.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116092679","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}
引用次数: 2
Rigidity results for Hermitian-Einstein manifolds 埃尔米特-爱因斯坦流形的刚性结果
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2013-11-25 DOI: 10.3318/PRIA.2016.116.03
Stuart J. Hall, T. Murphy
{"title":"Rigidity results for Hermitian-Einstein manifolds","authors":"Stuart J. Hall, T. Murphy","doi":"10.3318/PRIA.2016.116.03","DOIUrl":"https://doi.org/10.3318/PRIA.2016.116.03","url":null,"abstract":"A differential operator introduced by A. Gray on the unit sphere bundle of a K\"ahler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a K\"ahler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms amongst compact Hermitian surfaces and the product of two projective lines amongst all K\"ahler-Einstein surfaces are then derived.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"306 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116226224","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}
引用次数: 4
Non-commutative functional calculus in finite type I von Neumann algebras 有限型von Neumann代数中的非交换泛函演算
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2013-10-21 DOI: 10.3318/pria.2016.116.10
Piotr Niemiec
{"title":"Non-commutative functional calculus in finite type I von Neumann algebras","authors":"Piotr Niemiec","doi":"10.3318/pria.2016.116.10","DOIUrl":"https://doi.org/10.3318/pria.2016.116.10","url":null,"abstract":"A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116528849","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}
引用次数: 0
Wiman-Valiron theory for a class of functions meromorphic in the unit disc 单位圆盘上一类亚纯函数的Wiman-Valiron理论
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2013-09-04 DOI: 10.3318/PRIA.2014.114.14
J. Langley, J. Rossi
{"title":"Wiman-Valiron theory for a class of functions meromorphic in the unit disc","authors":"J. Langley, J. Rossi","doi":"10.3318/PRIA.2014.114.14","DOIUrl":"https://doi.org/10.3318/PRIA.2014.114.14","url":null,"abstract":"Analogues of the key results of Wiman-Valiron theory are proved for a class of functions meromorphic in the unit disc, based on an approach developed by Bergweiler, Rippon and Stallard for the plane setting. The results give local approximations for the function and its logarithmic derivative and, in the case of positive order of growth, for higher order logarithmic derivatives as well.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128900825","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}
引用次数: 0
Multiplicative groups of division rings 除法环的乘法群
Mathematical Proceedings of the Royal Irish Academy Pub Date : 2013-02-04 DOI: 10.3318/PRIA.2014.114.04
R. Hazrat, M. Mahdavi-Hezavehi, M. Motiee
{"title":"Multiplicative groups of division rings","authors":"R. Hazrat, M. Mahdavi-Hezavehi, M. Motiee","doi":"10.3318/PRIA.2014.114.04","DOIUrl":"https://doi.org/10.3318/PRIA.2014.114.04","url":null,"abstract":"Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126355446","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}
引用次数: 13
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