Gerardo M. Escolano, Antonio M. Peralta, Armando R. Villena
Let $mathfrak{M}$ and $mathfrak{J}$ be JBW$^*$-algebras admitting no central summands of type $I_1$ and $I_2,$ and let $Phi: mathfrak{M} rightarrow mathfrak{J}$ be a linear bijection preserving operator commutativity in both directions, that is, $$[x,mathfrak{M},y] = 0 Leftrightarrow [Phi(x),mathfrak{J},Phi(y)] = 0,$$ for all $x,yin mathfrak{M}$, where the associator of three elements $a,b,c$ in $mathfrak{M}$ is defined by $[a,b,c]:=(acirc b)circ c - (ccirc b)circ a$. We prove that under these conditions there exist a unique invertible central element $z_0$ in $mathfrak{J}$, a unique Jordan isomorphism $J: mathfrak{M} rightarrow mathfrak{J}$, and a unique linear mapping $beta$ from $mathfrak{M}$ to the centre of $mathfrak{J}$ satisfying $$ Phi(x) = z_0 circ J(x) + beta(x), $$ for all $xin mathfrak{M}.$ Furthermore, if $Phi$ is a symmetric mapping (i.e., $Phi (x^*) = Phi (x)^*$ for all $xin mathfrak{M}$), the element $z_0$ is self-adjoint, $J$ is a Jordan $^*$-isomorphism, and $beta$ is a symmetric mapping too. In case that $mathfrak{J}$ is a JBW$^*$-algebra admitting no central summands of type $I_1$, we also address the problem of describing the form of all symmetric bilinear mappings $B : mathfrak{J}times mathfrak{J}to mathfrak{J}$ whose trace is associating (i.e., $[B(a,a),b,a] = 0,$ for all $a, b in mathfrak{J})$ providing a complete solution to it. We also determine the form of all associating linear maps on $mathfrak{J}$.
{"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}
In the theory of C*-algebras, the Weyl groups were defined for the Cuntz algebras and graph algebras by Cuntz and Conti et.al respectively. In this paper, we introduce and investigate the Weyl groups of groupoid C*-algebras as a natural generalization of the existing Weyl groups. Then we analyse several groups of automorphisms on groupoid C*-algebras. Finally, we apply our results to Cuntz algebras, graph algebras and C*-algebras associated with Deaconu-Renault systems.
{"title":"Weyl groups of groupoid C*-algebras","authors":"Fuyuta Komura","doi":"arxiv-2409.04906","DOIUrl":"https://doi.org/arxiv-2409.04906","url":null,"abstract":"In the theory of C*-algebras, the Weyl groups were defined for the Cuntz\u0000algebras and graph algebras by Cuntz and Conti et.al respectively. In this\u0000paper, we introduce and investigate the Weyl groups of groupoid C*-algebras as\u0000a natural generalization of the existing Weyl groups. Then we analyse several\u0000groups of automorphisms on groupoid C*-algebras. Finally, we apply our results\u0000to Cuntz algebras, graph algebras and C*-algebras associated with\u0000Deaconu-Renault systems.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195016","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 purpose of this paper is to characterize several classes of functional identities involving inverses with related mappings from a unital Banach algebra $mathcal{A}$ over the complex field into a unital $mathcal{A}$-bimodule $mathcal{M}$. Let $N$ be a fixed invertible element in $mathcal{A}$, $M$ be a fixed element in $mathcal{M}$, and $n$ be a positive integer. We investigate the forms of additive mappings $f$, $g$ from $mathcal{A}$ into $mathcal{M}$ satisfying one of the following identities: begin{equation*} begin{aligned} &f(A)A- Ag(A) = 0 &f(A)+ g(B)star A= M &f(A)+A^{n}g(A^{-1})=0 &f(A)+A^{n}g(B)=M end{aligned} qquad begin{aligned} &text{for each invertible element}~Ainmathcal{A}; &text{whenever}~ A,Binmathcal{A}~text{with}~AB=N; &text{for each invertible element}~Ainmathcal{A}; &text{whenever}~ A,Binmathcal{A}~text{with}~AB=N, end{aligned} end{equation*} where $star$ is either the Jordan product $Astar B = AB+BA$ or the Lie product $Astar B = AB-BA$.
{"title":"Functional identities involving inverses on Banach algebras","authors":"Kaijia Luo, Jiankui Li","doi":"arxiv-2409.04192","DOIUrl":"https://doi.org/arxiv-2409.04192","url":null,"abstract":"The purpose of this paper is to characterize several classes of functional\u0000identities involving inverses with related mappings from a unital Banach\u0000algebra $mathcal{A}$ over the complex field into a unital\u0000$mathcal{A}$-bimodule $mathcal{M}$. Let $N$ be a fixed invertible element in\u0000$mathcal{A}$, $M$ be a fixed element in $mathcal{M}$, and $n$ be a positive\u0000integer. We investigate the forms of additive mappings $f$, $g$ from\u0000$mathcal{A}$ into $mathcal{M}$ satisfying one of the following identities:\u0000begin{equation*} begin{aligned} &f(A)A- Ag(A) = 0 &f(A)+ g(B)star A= M\u0000&f(A)+A^{n}g(A^{-1})=0 &f(A)+A^{n}g(B)=M end{aligned} qquad begin{aligned}\u0000&text{for each invertible element}~Ainmathcal{A}; &text{whenever}~\u0000A,Binmathcal{A}~text{with}~AB=N; &text{for each invertible\u0000element}~Ainmathcal{A}; &text{whenever}~\u0000A,Binmathcal{A}~text{with}~AB=N, end{aligned} end{equation*} where $star$\u0000is either the Jordan product $Astar B = AB+BA$ or the Lie product $Astar B =\u0000AB-BA$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195018","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}
Pierre Clare, Chi-Kwong Li, Edward Poon, Eric Swartz
We study the problem of calculating noncommutative distances on graphs, using techniques from linear algebra, specifically, Birkhoff-James orthogonality. A complete characterization of the solutions is obtained in the case when the underlying graph is a path.
{"title":"Noncommutative distances on graphs: An explicit approach via Birkhoff-James orthogonality","authors":"Pierre Clare, Chi-Kwong Li, Edward Poon, Eric Swartz","doi":"arxiv-2409.04146","DOIUrl":"https://doi.org/arxiv-2409.04146","url":null,"abstract":"We study the problem of calculating noncommutative distances on graphs, using\u0000techniques from linear algebra, specifically, Birkhoff-James orthogonality. A\u0000complete characterization of the solutions is obtained in the case when the\u0000underlying graph is a path.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195036","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}
Aluthge transform is a well-known mapping defined on bounded linear operators. Especially, the convergence property of its iteration has been studied by many authors. In this paper, we discuss the problem for the induced Aluthge transforms which is a generalization of the Aluthge transform defined in 2021. We give the polar decomposition of the induced Aluthge transformations of centered operators and show its iteration converges to a normal operator. In particular, if $T$ is an invertible centered matrix, then iteration of any induced Aluthge transformations converges. Using the canonical standard form of matrix algebras we show that the iteration of any induced Aluthge transformations with respect to the weighted arithmetic mean and the power mean converge. Those observation are extended to the $C^*$-algebra of compact operators on an infinite dimensional Hilbert space, and as an application we show the stability of $mathcal{AN}$ and $mathcal{AM}$ properties under the iteration of the induced Aluthge transformations. We also provide concrete forms of their limit points for centered matrices and several examples. Moreover, we discuss the limit point of the induced Aluthge transformation with respect to the power mean in the injective $II_1$-factor $mathcal{M}$ and determine the form of its limit for some centered operators in $mathcal{M}$.
{"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}
We prove the existence of a C*-diagonal in the Cuntz algebra O_2 with spectrum homeomorphic to the Cantor space.
我们证明了 Cuntz 代数 O_2 中存在一个 C* 对角线，其谱与康托尔空间同构。
{"title":"A Cantor spectrum diagonal in O_2","authors":"Philipp Sibbel, Wilhelm Winter","doi":"arxiv-2409.03511","DOIUrl":"https://doi.org/arxiv-2409.03511","url":null,"abstract":"We prove the existence of a C*-diagonal in the Cuntz algebra O_2 with\u0000spectrum homeomorphic to the Cantor space.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195021","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 prove that (a) the sections space of a continuous unital subhomogeneous $C^*$ bundle over compact metrizable $X$ admits a finite-index expectation onto $C(X)$, answering a question of Blanchard-Gogi'{c} (in the metrizable case); (b) such expectations cannot, generally, have ``optimal index'', answering negatively a variant of the same question; and (c) a homogeneous continuous Banach bundle over a locally paracompact base space $X$ can be renormed into a Hilbert bundle in such a manner that the original space of bounded sections is $C_b(X)$-linearly Banach-Mazur-close to the resulting Hilbert module over the algebra $C_b(X)$ of continuous bounded functions on $X$. This last result resolves quantitatively another problem posed by Gogi'{c}.
{"title":"Non-commutative branched covers and bundle unitarizability","authors":"Alexandru Chirvasitu","doi":"arxiv-2409.03531","DOIUrl":"https://doi.org/arxiv-2409.03531","url":null,"abstract":"We prove that (a) the sections space of a continuous unital subhomogeneous\u0000$C^*$ bundle over compact metrizable $X$ admits a finite-index expectation onto\u0000$C(X)$, answering a question of Blanchard-Gogi'{c} (in the metrizable case);\u0000(b) such expectations cannot, generally, have ``optimal index'', answering\u0000negatively a variant of the same question; and (c) a homogeneous continuous\u0000Banach bundle over a locally paracompact base space $X$ can be renormed into a\u0000Hilbert bundle in such a manner that the original space of bounded sections is\u0000$C_b(X)$-linearly Banach-Mazur-close to the resulting Hilbert module over the\u0000algebra $C_b(X)$ of continuous bounded functions on $X$. This last result\u0000resolves quantitatively another problem posed by Gogi'{c}.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195050","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 order to circumvent a fundamental issue when studying densely defined traces on $mathrm{C}^ast$-algebras -- which we refer to as the Trace Question -- we initiate a systematic study of the set $T_{mathbb R}(A)$ of self-adjoint traces on the Pedersen ideal of $A$. The set $T_{mathbb R}(A)$ is a topological vector space with a vector lattice structure, which in the unital setting reflects the Choquet simplex structure of the tracial states. We establish a form of Kadison duality for $T_{mathbb R}(A)$ and compute $T_{mathbb R}(A)$ for principal twisted 'etale groupoid $mathrm{C}^ast$-algebras. We also answer the Trace Question positively for a large class of $mathrm{C}^ast$-algebras.
{"title":"Self-adjoint traces on the Pedersen ideal of $mathrm{C}^ast$-algebras","authors":"James Gabe, Alistair Miller","doi":"arxiv-2409.03587","DOIUrl":"https://doi.org/arxiv-2409.03587","url":null,"abstract":"In order to circumvent a fundamental issue when studying densely defined\u0000traces on $mathrm{C}^ast$-algebras -- which we refer to as the Trace Question\u0000-- we initiate a systematic study of the set $T_{mathbb R}(A)$ of self-adjoint\u0000traces on the Pedersen ideal of $A$. The set $T_{mathbb R}(A)$ is a topological vector space with a vector\u0000lattice structure, which in the unital setting reflects the Choquet simplex\u0000structure of the tracial states. We establish a form of Kadison duality for\u0000$T_{mathbb R}(A)$ and compute $T_{mathbb R}(A)$ for principal twisted 'etale\u0000groupoid $mathrm{C}^ast$-algebras. We also answer the Trace Question\u0000positively for a large class of $mathrm{C}^ast$-algebras.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195020","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}
Nekrashevych associated to each self-similar group action an ample groupoid and a C*-algebra. We provide exact sequences to compute the homology of the groupoid and the K-theory of the C*-algebra in terms of the homology of the group and K-theory of the group C*-algebra via the transfer map and the virtual endomorphism. Complete computations are then performed for the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups and many others. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of Xin Li, we are able to show that R"over's simple group containing the Grigorchuk group is rationally acyclic but has nontrivial Schur multiplier. We prove many more R"over--Nekrashevych groups of self-similar groups are rationally acyclic.
内克拉舍维奇（Nekrashevych）为每个自相似群作用关联了一个充裕群和一个 C* 代数。我们提供了精确的序列，通过转移映射和虚内变，以群的同源性和群 C* 代数的 K 理论来计算群的同源性和 C* 代数的 K 理论。然后对格里高丘克群、格里高丘克--埃尔斯克勒群、古普塔--西斯基群等进行了完整的计算。由于我们的结果和李昕最近的结果，我们能够证明包含格里高丘克群的R/"over'simple群是有理无循环的，但没有琐碎的舒尔乘数。我们还证明了更多自相似群的R（over--Nekrashevych）群是合理无循环的。
{"title":"Homology and K-theory for self-similar actions of groups and groupoids","authors":"Alistair Miller, Benjamin Steinberg","doi":"arxiv-2409.02359","DOIUrl":"https://doi.org/arxiv-2409.02359","url":null,"abstract":"Nekrashevych associated to each self-similar group action an ample groupoid\u0000and a C*-algebra. We provide exact sequences to compute the homology of the\u0000groupoid and the K-theory of the C*-algebra in terms of the homology of the\u0000group and K-theory of the group C*-algebra via the transfer map and the virtual\u0000endomorphism. Complete computations are then performed for the Grigorchuk\u0000group, the Grigorchuk--Erschler group, Gupta--Sidki groups and many others.\u0000Results are proved more generally for self-similar groupoids. As a consequence\u0000of our results and recent results of Xin Li, we are able to show that R\"over's\u0000simple group containing the Grigorchuk group is rationally acyclic but has\u0000nontrivial Schur multiplier. We prove many more R\"over--Nekrashevych groups of\u0000self-similar groups are rationally acyclic.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195032","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}
Consider a countably generated Hilbert $C^*$-module $mathcal M$ over a $C^*$-algebra $mathcal A$. There is a measure of noncompactness $lambda$ defined, roughly as the distance from finitely generated projective submodules, which is independent of any topology. We compare $lambda$ to the Hausdorff measure of noncompactness with respect to the family of seminorms that induce a topology recently iontroduced by Troitsky, denoted by $chi^*$. We obtain $lambdaequivchi^*$. Related inequalities involving other known measures of noncompactness, e.g. Kuratowski and Istru{a}c{t}escu are laso obtained as well as some related results on adjontable operators.
{"title":"Measures of noncompactness in Hilbert $C^*$-modules","authors":"Dragoljub J. Kečkić, Zlatko Lazović","doi":"arxiv-2409.02514","DOIUrl":"https://doi.org/arxiv-2409.02514","url":null,"abstract":"Consider a countably generated Hilbert $C^*$-module $mathcal M$ over a\u0000$C^*$-algebra $mathcal A$. There is a measure of noncompactness $lambda$\u0000defined, roughly as the distance from finitely generated projective submodules,\u0000which is independent of any topology. We compare $lambda$ to the Hausdorff\u0000measure of noncompactness with respect to the family of seminorms that induce a\u0000topology recently iontroduced by Troitsky, denoted by $chi^*$. We obtain\u0000$lambdaequivchi^*$. Related inequalities involving other known measures of\u0000noncompactness, e.g. Kuratowski and Istru{a}c{t}escu are laso obtained as\u0000well as some related results on adjontable operators.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195031","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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