Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at...Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the ease of Hopf C^＊-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C^＊-algebra are proposed.展开更多
Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Ge...Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.展开更多
In this article,we introduce and study the class of approximately Artinian(Noetherian)C^(*)-algebras,called AR-algebras(AN-algebras),which is a simultaneous generalization of Artinian(Noetherian)C*-algebras and AF-alg...In this article,we introduce and study the class of approximately Artinian(Noetherian)C^(*)-algebras,called AR-algebras(AN-algebras),which is a simultaneous generalization of Artinian(Noetherian)C*-algebras and AF-algebras.We study properties such as the ideal property and topological dimension zero for them.In particular,we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite.This extends the Kirchberg's O_(∞)-absorption theorem,and implies that a weakly purely infinite C^(*)-algebra is Noetherian iff every its ideal has a full projection.展开更多
Generally,the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dyn...Generally,the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models.C^(∗)-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research.The concept of a C^(∗)-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space.In fact,It is a generalization by replacing the set of real numbers with a C^(∗)-algebra.After that,this line of research continued,where several fixed point results have been obtained in the framework of C^(∗)-algebra valued metric,aswell as(more general)C^(∗)-algebra-valued b-metric spaces andC^(∗)-algebra-valued extended b-metric spaces.Very recently,based on the concept and properties of C^(∗)-algebras,we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case.In this paper,we first introduce the concept of C^(∗)-algebra-valued quasi-controlledK-metric spaces and prove some fixed point theorems that remain valid in this setting.To support our main results,we also furnish some exampleswhichdemonstrate theutility of ourmainresult.Finally,as an application,we useour results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.展开更多
In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some ...In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.展开更多
文摘Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the ease of Hopf C^＊-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C^＊-algebra are proposed.
文摘Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.
基金jointly supported by the National Natural Science Foundation of China(No.22273093,No.41905018,No.21903080)the Ministry of Science and Technology of China(No.2022YFF0606500)。
基金supported by grants from INSF(98029498,99013953)partly supported by a grant from IPM(96430215)。
文摘In this article,we introduce and study the class of approximately Artinian(Noetherian)C^(*)-algebras,called AR-algebras(AN-algebras),which is a simultaneous generalization of Artinian(Noetherian)C*-algebras and AF-algebras.We study properties such as the ideal property and topological dimension zero for them.In particular,we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite.This extends the Kirchberg's O_(∞)-absorption theorem,and implies that a weakly purely infinite C^(*)-algebra is Noetherian iff every its ideal has a full projection.
文摘Generally,the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models.C^(∗)-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research.The concept of a C^(∗)-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space.In fact,It is a generalization by replacing the set of real numbers with a C^(∗)-algebra.After that,this line of research continued,where several fixed point results have been obtained in the framework of C^(∗)-algebra valued metric,aswell as(more general)C^(∗)-algebra-valued b-metric spaces andC^(∗)-algebra-valued extended b-metric spaces.Very recently,based on the concept and properties of C^(∗)-algebras,we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case.In this paper,we first introduce the concept of C^(∗)-algebra-valued quasi-controlledK-metric spaces and prove some fixed point theorems that remain valid in this setting.To support our main results,we also furnish some exampleswhichdemonstrate theutility of ourmainresult.Finally,as an application,we useour results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.
基金Supported in part by the National Social Science Foundation of China(19BTJ020)。
文摘In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.
