arXiv - MATH - Operator Algebras最新文献_第2页

Preservers of Operator Commutativity 算子交换性的保全器

arXiv - MATH - Operator Algebras Pub Date : 2024-09-10 DOI: arxiv-2409.06799

Gerardo M. Escolano, Antonio M. Peralta, Armando R. Villena

{"title":"Preservers of Operator Commutativity","authors":"Gerardo M. Escolano, Antonio M. Peralta, Armando R. Villena","doi":"arxiv-2409.06799","DOIUrl":"https://doi.org/arxiv-2409.06799","url":null,"abstract":"Let $mathfrak{M}$ and $mathfrak{J}$ be JBW$^*$-algebras admitting no\u0000central summands of type $I_1$ and $I_2,$ and let $Phi: mathfrak{M}\u0000rightarrow mathfrak{J}$ be a linear bijection preserving operator\u0000commutativity in both directions, that is, $$[x,mathfrak{M},y] = 0\u0000Leftrightarrow [Phi(x),mathfrak{J},Phi(y)] = 0,$$ for all $x,yin\u0000mathfrak{M}$, where the associator of three elements $a,b,c$ in $mathfrak{M}$\u0000is defined by $[a,b,c]:=(acirc b)circ c - (ccirc b)circ a$. We prove that\u0000under these conditions there exist a unique invertible central element $z_0$ in\u0000$mathfrak{J}$, a unique Jordan isomorphism $J: mathfrak{M} rightarrow\u0000mathfrak{J}$, and a unique linear mapping $beta$ from $mathfrak{M}$ to the\u0000centre of $mathfrak{J}$ satisfying $$ Phi(x) = z_0 circ J(x) + beta(x), $$\u0000for all $xin mathfrak{M}.$ Furthermore, if $Phi$ is a symmetric mapping\u0000(i.e., $Phi (x^*) = Phi (x)^*$ for all $xin mathfrak{M}$), the element\u0000$z_0$ is self-adjoint, $J$ is a Jordan $^*$-isomorphism, and $beta$ is a\u0000symmetric mapping too. In case that $mathfrak{J}$ is a JBW$^*$-algebra admitting no central\u0000summands of type $I_1$, we also address the problem of describing the form of\u0000all symmetric bilinear mappings $B : mathfrak{J}times mathfrak{J}to\u0000mathfrak{J}$ whose trace is associating (i.e., $[B(a,a),b,a] = 0,$ for all $a,\u0000b in mathfrak{J})$ providing a complete solution to it. We also determine the\u0000form of all associating linear maps on $mathfrak{J}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195019","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

Weyl groups of groupoid C*-algebras 类群 C* 结构的韦尔群

arXiv - MATH - Operator Algebras Pub Date : 2024-09-07 DOI: arxiv-2409.04906

Fuyuta Komura

引用次数: 0

Functional identities involving inverses on Banach algebras 涉及巴拿赫数列上倒数的函数等式

arXiv - MATH - Operator Algebras Pub Date : 2024-09-06 DOI: arxiv-2409.04192

Kaijia Luo, Jiankui Li

引用次数: 0

Noncommutative distances on graphs: An explicit approach via Birkhoff-James orthogonality 图上的非交换距离：通过伯克霍夫-詹姆斯正交性的显式方法

arXiv - MATH - Operator Algebras Pub Date : 2024-09-06 DOI: arxiv-2409.04146

Pierre Clare, Chi-Kwong Li, Edward Poon, Eric Swartz

引用次数: 0

Limit of iteration of the induced Aluthge transformations of centered operators 居中算子的诱导阿卢斯格变换的迭代极限

arXiv - MATH - Operator Algebras Pub Date : 2024-09-05 DOI: arxiv-2409.03338

Hiroyuki Osaka, Takeaki Yamazaki

{"title":"Limit of iteration of the induced Aluthge transformations of centered operators","authors":"Hiroyuki Osaka, Takeaki Yamazaki","doi":"arxiv-2409.03338","DOIUrl":"https://doi.org/arxiv-2409.03338","url":null,"abstract":"Aluthge transform is a well-known mapping defined on bounded linear\u0000operators. Especially, the convergence property of its iteration has been\u0000studied by many authors. In this paper, we discuss the problem for the induced\u0000Aluthge transforms which is a generalization of the Aluthge transform defined\u0000in 2021. We give the polar decomposition of the induced Aluthge transformations\u0000of centered operators and show its iteration converges to a normal operator. In\u0000particular, if $T$ is an invertible centered matrix, then iteration of any\u0000induced Aluthge transformations converges. Using the canonical standard form of\u0000matrix algebras we show that the iteration of any induced Aluthge\u0000transformations with respect to the weighted arithmetic mean and the power mean\u0000converge. Those observation are extended to the $C^*$-algebra of compact\u0000operators on an infinite dimensional Hilbert space, and as an application we\u0000show the stability of $mathcal{AN}$ and $mathcal{AM}$ properties under the\u0000iteration of the induced Aluthge transformations. We also provide concrete\u0000forms of their limit points for centered matrices and several examples.\u0000Moreover, we discuss the limit point of the induced Aluthge transformation with\u0000respect to the power mean in the injective $II_1$-factor $mathcal{M}$ and\u0000determine the form of its limit for some centered operators in $mathcal{M}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195029","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

A Cantor spectrum diagonal in O_2 O_2 中对角线的康托谱

arXiv - MATH - Operator Algebras Pub Date : 2024-09-05 DOI: arxiv-2409.03511

Philipp Sibbel, Wilhelm Winter

引用次数: 0

Non-commutative branched covers and bundle unitarizability 非交换支盖和束单可化性

arXiv - MATH - Operator Algebras Pub Date : 2024-09-05 DOI: arxiv-2409.03531

Alexandru Chirvasitu

引用次数: 0

Self-adjoint traces on the Pedersen ideal of $mathrm{C}^ast$-algebras $mathrm{C}^ast$-原子的佩德森理想上的自共迹

arXiv - MATH - Operator Algebras Pub Date : 2024-09-05 DOI: arxiv-2409.03587

James Gabe, Alistair Miller