We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by $C^*$-algebras and inspired by the realization of the K-theory of a $C^*$-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the $K_0$-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For $C^*$-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.
我们提议将 K 理论推广到算子系统。受$C^*$-代数描述的非交换空间的谱截断的启发，以及将$C^*$-代数的K理论实现为赫米特形式的维特群的启发，我们引入了以相应矩阵大小为索引的新的算子系统不变式。我们构建了一个直接系统，它的直接极限具有半群结构，我们将 $K_0$ 群定义为相应的格罗内狄克群。这是单元算子系统的不变式，更一般地说，是算子系统的莫里塔等价不变式。对于$C^*$数组，它可以简化为通常的定义。我们通过谱定位器来说明我们的不变量。
{"title":"A generalization of K-theory to operator systems","authors":"Walter D. van Suijlekom","doi":"arxiv-2409.02773","DOIUrl":"https://doi.org/arxiv-2409.02773","url":null,"abstract":"We propose a generalization of K-theory to operator systems. Motivated by\u0000spectral truncations of noncommutative spaces described by $C^*$-algebras and\u0000inspired by the realization of the K-theory of a $C^*$-algebra as the Witt\u0000group of hermitian forms, we introduce new operator system invariants indexed\u0000by the corresponding matrix size. A direct system is constructed whose direct\u0000limit possesses a semigroup structure, and we define the $K_0$-group as the\u0000corresponding Grothendieck group. This is an invariant of unital operator\u0000systems, and, more generally, an invariant up to Morita equivalence of operator\u0000systems. For $C^*$-algebras it reduces to the usual definition. We illustrate\u0000our invariant by means of the spectral localizer.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195030","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 article, we identify a suitable approach to define the character space of a commutative unital locally $C^{ast}$-algebra via the notion of the inductive limit of topological spaces. Also, we discuss topological properties of the character space. We establish the Gelfand type representation between a commutative unital locally $C^{ast}$-algebra and the space of all continuous functions defined on its character space. Equivalently, we prove that every commutative unital locally $C^{ast}$-algebra is identified with the locally $C^{ast}$-algebra of continuous functions on its character space through the coherent representation of projective limit of $C^{ast}$-algebras. Finally, we construct a unital locally $C^{ast}$-algebra generated by a given locally bounded normal operator and show that its character space is homeomorphic to the local spectrum. Further, we define the functional calculus and prove spectral mapping theorem in this framework.
{"title":"Character Space and Gelfand type representation of locally C^{*}-algebra","authors":"Santhosh Kumar Pamula, Rifat Siddique","doi":"arxiv-2409.01755","DOIUrl":"https://doi.org/arxiv-2409.01755","url":null,"abstract":"In this article, we identify a suitable approach to define the character\u0000space of a commutative unital locally $C^{ast}$-algebra via the notion of the\u0000inductive limit of topological spaces. Also, we discuss topological properties\u0000of the character space. We establish the Gelfand type representation between a\u0000commutative unital locally $C^{ast}$-algebra and the space of all continuous\u0000functions defined on its character space. Equivalently, we prove that every\u0000commutative unital locally $C^{ast}$-algebra is identified with the locally\u0000$C^{ast}$-algebra of continuous functions on its character space through the\u0000coherent representation of projective limit of $C^{ast}$-algebras. Finally, we\u0000construct a unital locally $C^{ast}$-algebra generated by a given locally\u0000bounded normal operator and show that its character space is homeomorphic to\u0000the local spectrum. Further, we define the functional calculus and prove\u0000spectral mapping theorem in this framework.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195033","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 work, we introduce the concept of direct integral of locally Hilbert spaces by using the notion of locally standard measure space (analogous to standard measure space defined in the classical setup), which we obtain by considering a strictly inductive system of measurable spaces along with a projective system of finite measures. Next, we define a locally Hilbert space given by the direct integral of a family of locally Hilbert spaces. Following that we introduce decomposable locally bounded and diagonalizable locally bounded operators. Further, we show that the class of diagonalizable locally bounded operators is an abelian locally von Neumann algebra, and this can be seen as the commutant of decomposable locally bounded operators. Finally, we discuss the following converse question: For a locally Hilbert space $mathcal{D}$ and an abelian locally von Neumann algebra $mathcal{M}$, does there exist a locally standard measure space and a family of locally Hilbert spaces such that (1) the locally Hilbert space $mathcal{D}$ is identified with the direct integral of family of locally Hilbert spaces; (2) the abelian locally von Neumann algebra $mathcal{M}$ is identified with the abelian locally von Neumann algebra of all diagonalizable locally bounded operators ? We answer this question affirmatively for a certain class of abelian locally von Neumann algebras.
{"title":"Direct Integral and Decompoisitions of Locally Hilbert spaces","authors":"Chaitanya J. Kulkarni, Santhosh Kumar Pamula","doi":"arxiv-2409.01200","DOIUrl":"https://doi.org/arxiv-2409.01200","url":null,"abstract":"In this work, we introduce the concept of direct integral of locally Hilbert\u0000spaces by using the notion of locally standard measure space (analogous to\u0000standard measure space defined in the classical setup), which we obtain by\u0000considering a strictly inductive system of measurable spaces along with a\u0000projective system of finite measures. Next, we define a locally Hilbert space\u0000given by the direct integral of a family of locally Hilbert spaces. Following\u0000that we introduce decomposable locally bounded and diagonalizable locally\u0000bounded operators. Further, we show that the class of diagonalizable locally\u0000bounded operators is an abelian locally von Neumann algebra, and this can be\u0000seen as the commutant of decomposable locally bounded operators. Finally, we\u0000discuss the following converse question: For a locally Hilbert space $mathcal{D}$ and an abelian locally von Neumann\u0000algebra $mathcal{M}$, does there exist a locally standard measure space and a\u0000family of locally Hilbert spaces such that (1) the locally Hilbert space $mathcal{D}$ is identified with the direct\u0000integral of family of locally Hilbert spaces; (2) the abelian locally von Neumann algebra $mathcal{M}$ is identified with\u0000the abelian locally von Neumann algebra of all diagonalizable locally bounded\u0000operators ? We answer this question affirmatively for a certain class of abelian locally\u0000von Neumann algebras.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195034","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 quantum lens spaces form a natural and well-studied class of noncommutative spaces which has been partially classified using algebraic invariants drawing on the developed classification theory of graph $C^*$-algebras. We introduce the problem of deciding when two quantum lens spaces are equivariantly isomorphic, and solve it in certain basic cases. The results can be formulated directly in terms of the parameters defining the quantum lens spaces, and here occasionally take on a rather complicated from which convinces us that there is a deep underlying explanation for our findings. We complement the fully established partial results with computer experiments that may indicate the way forward.
{"title":"Equivariant isomorphism of Quantum Lens Spaces of low dimension","authors":"Søren Eilers, Sophie Emma Zegers","doi":"arxiv-2408.17386","DOIUrl":"https://doi.org/arxiv-2408.17386","url":null,"abstract":"The quantum lens spaces form a natural and well-studied class of\u0000noncommutative spaces which has been partially classified using algebraic\u0000invariants drawing on the developed classification theory of graph\u0000$C^*$-algebras. We introduce the problem of deciding when two quantum lens\u0000spaces are equivariantly isomorphic, and solve it in certain basic cases. The\u0000results can be formulated directly in terms of the parameters defining the\u0000quantum lens spaces, and here occasionally take on a rather complicated from\u0000which convinces us that there is a deep underlying explanation for our\u0000findings. We complement the fully established partial results with computer\u0000experiments that may indicate the way forward.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195035","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 investigate the relationship between ideal structures and the Bockstein operations in the total K-theory, offering various diagrams to demonstrate their effectiveness in classification. We explore different situations and demonstrate a variety of conclusions, highlighting the crucial role these structures play within the framework of invariants.
在本文中，我们研究了理想结构与全 K 理论中的博克斯坦运算之间的关系，并提供了各种图表来证明它们在分类中的有效性。我们探讨了不同的情况，证明了各种结论，突出了这些结构在不变式框架中的关键作用。
{"title":"Bockstein operations and extensions with trivial boundary maps","authors":"Qingnan An, Zhichao Liu","doi":"arxiv-2408.17055","DOIUrl":"https://doi.org/arxiv-2408.17055","url":null,"abstract":"In this paper, we investigate the relationship between ideal structures and\u0000the Bockstein operations in the total K-theory, offering various diagrams to\u0000demonstrate their effectiveness in classification. We explore different\u0000situations and demonstrate a variety of conclusions, highlighting the crucial\u0000role these structures play within the framework of invariants.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195037","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 $Omega$ be a compact subset of $mathbb{C}$ and let $A$ be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $alpha:{rm Cu}(C(Omega))to {rm Cu}(A)$ with $alpha(langle mathds{1}_{Omega}rangle)leq langle 1_Arangle$, there exists a homomorphism $phi: C(Omega)to A$ such that ${rm Cu}(phi)=alpha$ and $phi$ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$-algebras to $A$ in terms of the Cuntz semigroup.
{"title":"Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra","authors":"Qingnan An, George Elliott, Zhichao Liu","doi":"arxiv-2408.16657","DOIUrl":"https://doi.org/arxiv-2408.16657","url":null,"abstract":"Let $Omega$ be a compact subset of $mathbb{C}$ and let $A$ be a unital\u0000simple, separable $C^*$-algebra with stable rank one, real rank zero and strict\u0000comparison. We show that, given a Cu-morphism $alpha:{rm Cu}(C(Omega))to\u0000{rm Cu}(A)$ with $alpha(langle mathds{1}_{Omega}rangle)leq langle\u00001_Arangle$, there exists a homomorphism $phi: C(Omega)to A$ such that ${rm\u0000Cu}(phi)=alpha$ and $phi$ is unique up to approximate unitary equivalence.\u0000We also give classification results for maps from a large class of\u0000$C^*$-algebras to $A$ in terms of the Cuntz semigroup.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195039","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}
David Gao, Greg Patchell, Srivatsav Kunnawalkam Elayavalli
We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is preserved by taking several natural graph join operations. We prove that an action of a group on its Cayley graph is sofic if and only if the group is sofic. We show that arbitrary actions of amenable groups on graphs are sofic. Using a graph theoretic result of E. Hrushovski, we also show that arbitrary actions of free groups on graphs are sofic. Notably we show that arbitrary actions of sofic groups on graphs, with amenable stabilizers, are sofic, settling completely an open problem from cite{gao2024soficity}. We also show that soficity is preserved by taking limits under a natural Gromov-Hausdorff topology, generalizing prior work of the first author cite{gao2024actionslerfgroupssets}. Our work sheds light on a family of groups called graph wreath products, simultaneously generalizing graph products and generalized wreath products. Extending various prior results in this direction including soficity of generalized wreath products cite{gao2024soficity}, B. Hayes and A. Sale cite{HayesSale}, and soficity of graph products cite{CHR, charlesworth2021matrix}, we show that graph wreath products are sofic if the action and acting groups are sofic. These results provide several new examples of sofic groups in a systematic manner.
{"title":"Sofic actions on graphs","authors":"David Gao, Greg Patchell, Srivatsav Kunnawalkam Elayavalli","doi":"arxiv-2408.15470","DOIUrl":"https://doi.org/arxiv-2408.15470","url":null,"abstract":"We develop a theory of soficity for actions on graphs and obtain new\u0000applications to the study of sofic groups. We establish various examples,\u0000stability and permanence properties of sofic actions on graphs, in particular\u0000soficity is preserved by taking several natural graph join operations. We prove\u0000that an action of a group on its Cayley graph is sofic if and only if the group\u0000is sofic. We show that arbitrary actions of amenable groups on graphs are\u0000sofic. Using a graph theoretic result of E. Hrushovski, we also show that\u0000arbitrary actions of free groups on graphs are sofic. Notably we show that\u0000arbitrary actions of sofic groups on graphs, with amenable stabilizers, are\u0000sofic, settling completely an open problem from cite{gao2024soficity}. We also\u0000show that soficity is preserved by taking limits under a natural\u0000Gromov-Hausdorff topology, generalizing prior work of the first author\u0000cite{gao2024actionslerfgroupssets}. Our work sheds light on a family of groups\u0000called graph wreath products, simultaneously generalizing graph products and\u0000generalized wreath products. Extending various prior results in this direction\u0000including soficity of generalized wreath products cite{gao2024soficity}, B.\u0000Hayes and A. Sale cite{HayesSale}, and soficity of graph products cite{CHR,\u0000charlesworth2021matrix}, we show that graph wreath products are sofic if the\u0000action and acting groups are sofic. These results provide several new examples\u0000of sofic groups in a systematic manner.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195040","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, a new invariant was built towards the classification of separable C*-algebras of real rank zero, which we call latticed total K-theory. A classification theorem is given in terms of such an invariant for a large class of separable C*-algebras of real rank zero arising from the extensions of finite and infinite C*-algebras. Many algebras with both finite and infinite projections can be classified.
{"title":"A latticed total K-theory","authors":"Qingnan An, Chunguang Li, Zhichao Liu","doi":"arxiv-2408.15941","DOIUrl":"https://doi.org/arxiv-2408.15941","url":null,"abstract":"In this paper, a new invariant was built towards the classification of\u0000separable C*-algebras of real rank zero, which we call latticed total K-theory.\u0000A classification theorem is given in terms of such an invariant for a large\u0000class of separable C*-algebras of real rank zero arising from the extensions of\u0000finite and infinite C*-algebras. Many algebras with both finite and infinite\u0000projections can be classified.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195038","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 say that an inclusion of a $*$-algebra $A$ into a $C^*$-algebra $B$ has the ideal separation property if closed ideals in $B$ can be recovered by their intersection with $A$. Such inclusions have attractive properties from the point of view of harmonic analysis and noncommutative geometry. We establish several permanence properties of locally compact groups for which $L^1(G) subseteq C^*_{mathrm{red}}(G)$ has the ideal separation property.
{"title":"The ideal separation property for reduced group $C^*$-algebras","authors":"Are Austad, Hannes Thiel","doi":"arxiv-2408.14880","DOIUrl":"https://doi.org/arxiv-2408.14880","url":null,"abstract":"We say that an inclusion of a $*$-algebra $A$ into a $C^*$-algebra $B$ has\u0000the ideal separation property if closed ideals in $B$ can be recovered by their\u0000intersection with $A$. Such inclusions have attractive properties from the\u0000point of view of harmonic analysis and noncommutative geometry. We establish\u0000several permanence properties of locally compact groups for which $L^1(G)\u0000subseteq C^*_{mathrm{red}}(G)$ has the ideal separation property.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195041","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 a Hankel-variant commutant lifting theorem. This also uncovers the complete structure of the Beurling-type reducing and invariant subspaces of Hankel operators. Kernel spaces of Hankel operators play a key role in the analysis.
{"title":"Liftings and invariant subspaces of Hankel operators","authors":"Sneha B, Neeru Bala, Samir Panja, Jaydeb Sarkar","doi":"arxiv-2408.13753","DOIUrl":"https://doi.org/arxiv-2408.13753","url":null,"abstract":"We prove a Hankel-variant commutant lifting theorem. This also uncovers the\u0000complete structure of the Beurling-type reducing and invariant subspaces of\u0000Hankel operators. Kernel spaces of Hankel operators play a key role in the\u0000analysis.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195044","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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