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
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
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
We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-tr...We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.展开更多
Further to the functional representations of C^＊-algebras proposed by R. Cirelli and A. Manik, we consider the uniform Kahler bundle （UKB） description of some C^＊-algebraic subjects. In particular, we obtain a one...Further to the functional representations of C^＊-algebras proposed by R. Cirelli and A. Manik, we consider the uniform Kahler bundle （UKB） description of some C^＊-algebraic subjects. In particular, we obtain a one-to- one correspondence between closed ideals of a C^＊-algebra A and full uniform Kahler subbundles over open subsets of the base space of the UKB associated with A. In addition, we present a geometric description of the pure state space of hereditary C^＊-subalgebras and show that if B is a hereditary C^＊-subalgebra of A, the UKB of B is a kind of Kahler subbundle of the UKB of A. As a simple example, we consider hereditary C^＊-subalgebras of the C^＊-algebra of compact operators on a Hilbert space. Finally, we remark that each hereditary C^＊- subalgebra of A also can be naturally characterized as a uniform holomorphic Hilbert bundle.展开更多
We prove the Hyers-Ulam stability of linear N-isometries in linear N-normed Banach mod- ules over a unital C^＊-algebra. The main purpose of this paper is to investigate N-isometric C^＊-algebra isomorphisms between l...We prove the Hyers-Ulam stability of linear N-isometries in linear N-normed Banach mod- ules over a unital C^＊-algebra. The main purpose of this paper is to investigate N-isometric C^＊-algebra isomorphisms between linear N-normed C^＊-algebras, N-isometric Poisson C^＊-algebra isomorphisms between linear N-normed Poisson C^＊-algebras, N-isometric Lie C^＊-algebra isomorphisms between linear N-normed Lie C^＊-algebras, N-isometric Poisson JC^＊-algebra isomorphisms between linear N-normed Poisson JC^＊-algebras, and N-isometric Lie JC^＊-algebra isomorphisms between linear N-normed Lie JC^＊-algebras. Moreover, we prove the Hyers- Ulam stability of t：heir N-isometric homomorphisms.展开更多
Given two nuclear C^＊-algebras A1 and A2 with states φ1 and φ2, we show that the monotone product C^＊-algebra A1 △→ A2 is still nuclear. Furthermore, if both the states φ1 and φ2 are faithful, then the monoton...Given two nuclear C^＊-algebras A1 and A2 with states φ1 and φ2, we show that the monotone product C^＊-algebra A1 △→ A2 is still nuclear. Furthermore, if both the states φ1 and φ2 are faithful, then the monotone product ,A1 △→ A2 is nuclear if and only if the C^＊-algebras ,A1 and A2 both are nuclear.展开更多
We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a...We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C~*-algebra C~*(G) of G when G has property(RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G~0 of G, gives rise to a canonical map τon the algebra C(G) of complex continuous functions with compact support on G. We show that the map τcan be extended continuously to S~l(G) and plays the same role as an n-trace on C~*(G) when G has property(RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C~*(G).展开更多
它被看那从 pro-C 的连续线性地图的一个 n × n 矩阵[*]到 L (H)的代数学 A ，验证完全的确实的条件，具有形式[ V [*] T ij Φ(· ) V ]我， j=1 [ n ]，在Φ 是 Hilbert 空间 K 上的 A 的一个代表的地方， V 是从 H 的一...它被看那从 pro-C 的连续线性地图的一个 n × n 矩阵[*]到 L (H)的代数学 A ，验证完全的确实的条件，具有形式[ V [*] T ij Φ(· ) V ]我， j=1 [ n ]，在Φ 是 Hilbert 空间 K 上的 A 的一个代表的地方， V 是从 H 的一个围住的线性操作员到 K ，；[T ij ] n i， j=1 [n ] 是在 C 的一个积极元素[*] 在Φ(A 的 commutant 上的所有 n × n 矩阵的代数学) 在 L (K) 。这概括 C. Y 的结果。在 Proc 的 Suen。Amer。数学。Soc， 112 (3 ) ， 1991， 709 712。另外，这构造的一个 covariant 版本被给。展开更多
Concerning the stability problem of functional equations, we introduce a general （m, n）- Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general （m, n）-Cauchy Jensen ...Concerning the stability problem of functional equations, we introduce a general （m, n）- Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general （m, n）-Cauchy Jensen additive mappings in C^＊-algebras, which generalize the result＇s obtained for Cauchy-Jensen type additive mappings.展开更多
Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the(**)-Haagerup property for C*-algebras in this paper. They first give an a...Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the(**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki's question(2013), and then obtain several results of(**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the(**)-Haagerup property. Some heredity results concerning the(**)-Haagerup property are also proved.展开更多
目的为了提升生成对抗网络汉字风格迁移的图像生成质量,实现汉字智能生成在字库产业中的实际应用,提出了一种基于直观汉字构形学的条件生成对抗网络字体生成优化方法(Optimizationof Conditional Fonts Generation with Chinese Charact...目的为了提升生成对抗网络汉字风格迁移的图像生成质量,实现汉字智能生成在字库产业中的实际应用,提出了一种基于直观汉字构形学的条件生成对抗网络字体生成优化方法(Optimizationof Conditional Fonts Generation with Chinese Character Configuration GANs,C^(3)-GAN)。方法建构了直观汉字构形模组(C^(3)Module),该模组包含了利于条件生成对抗网络进行汉字构形语义特征学习的全特征汉字字符集。C^(3)-GAN在条件生成对抗网络模型下进行字体生成训练,降低了必要训练样本数量,实现对字体生成效果的优化。结果使用C^(3)-GAN生成汉字图像的清晰度更高、字形更准确。在图像相似性定量评估中,使用C^(3)-GAN的实验组相比于其他模型,获得了更高的相似值和更小的误差值。结论使用C^(3)-GAN可以降低必要训练样本数量、提升汉字图像质量。在实际项目中具有一定的应用性和可操作性。展开更多
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)。
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.
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.
基金I+D MEC Projects No.MTM 2005-02541,MTM 2004-03882Junta de Andalucfa Grants FQM 0199,FQM 0194,FQM 1215the PCI Project No.A/4044/05 of the Spanish AECI
文摘We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.
文摘Further to the functional representations of C^＊-algebras proposed by R. Cirelli and A. Manik, we consider the uniform Kahler bundle （UKB） description of some C^＊-algebraic subjects. In particular, we obtain a one-to- one correspondence between closed ideals of a C^＊-algebra A and full uniform Kahler subbundles over open subsets of the base space of the UKB associated with A. In addition, we present a geometric description of the pure state space of hereditary C^＊-subalgebras and show that if B is a hereditary C^＊-subalgebra of A, the UKB of B is a kind of Kahler subbundle of the UKB of A. As a simple example, we consider hereditary C^＊-subalgebras of the C^＊-algebra of compact operators on a Hilbert space. Finally, we remark that each hereditary C^＊- subalgebra of A also can be naturally characterized as a uniform holomorphic Hilbert bundle.
基金The first author is supported by Korea Research Foundation Grant KRF-2005-041-C00027
文摘We prove the Hyers-Ulam stability of linear N-isometries in linear N-normed Banach mod- ules over a unital C^＊-algebra. The main purpose of this paper is to investigate N-isometric C^＊-algebra isomorphisms between linear N-normed C^＊-algebras, N-isometric Poisson C^＊-algebra isomorphisms between linear N-normed Poisson C^＊-algebras, N-isometric Lie C^＊-algebra isomorphisms between linear N-normed Lie C^＊-algebras, N-isometric Poisson JC^＊-algebra isomorphisms between linear N-normed Poisson JC^＊-algebras, and N-isometric Lie JC^＊-algebra isomorphisms between linear N-normed Lie JC^＊-algebras. Moreover, we prove the Hyers- Ulam stability of t：heir N-isometric homomorphisms.
基金the Youth Foundation of Sichuan Education Department (No.2003B017)the Doctoral Foundation of Chongqing Normal University (No.08XLB013)
文摘Given two nuclear C^＊-algebras A1 and A2 with states φ1 and φ2, we show that the monotone product C^＊-algebra A1 △→ A2 is still nuclear. Furthermore, if both the states φ1 and φ2 are faithful, then the monotone product ,A1 △→ A2 is nuclear if and only if the C^＊-algebras ,A1 and A2 both are nuclear.
基金Supported by the NNSF of China(Grant Nos.11271224 and 11371290)
文摘We introduce the notion of property(RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S~l(G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C~*-algebra C~*(G) of G when G has property(RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G~0 of G, gives rise to a canonical map τon the algebra C(G) of complex continuous functions with compact support on G. We show that the map τcan be extended continuously to S~l(G) and plays the same role as an n-trace on C~*(G) when G has property(RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C~*(G).
基金Project supported by the grant CNCSIS (Romanian National Council for Research in High Education)-code A 1065/2006.
文摘它被看那从 pro-C 的连续线性地图的一个 n × n 矩阵[*]到 L (H)的代数学 A ，验证完全的确实的条件，具有形式[ V [*] T ij Φ(· ) V ]我， j=1 [ n ]，在Φ 是 Hilbert 空间 K 上的 A 的一个代表的地方， V 是从 H 的一个围住的线性操作员到 K ，；[T ij ] n i， j=1 [n ] 是在 C 的一个积极元素[*] 在Φ(A 的 commutant 上的所有 n × n 矩阵的代数学) 在 L (K) 。这概括 C. Y 的结果。在 Proc 的 Suen。Amer。数学。Soc， 112 (3 ) ， 1991， 709 712。另外，这构造的一个 covariant 版本被给。
文摘Concerning the stability problem of functional equations, we introduce a general （m, n）- Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general （m, n）-Cauchy Jensen additive mappings in C^＊-algebras, which generalize the result＇s obtained for Cauchy-Jensen type additive mappings.
基金supported by the National Natural Science Foundation of China(No.11371279)the Shandong Provincial Natural Science Foundation of China(No.ZR2015PA010)
文摘Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the(**)-Haagerup property for C*-algebras in this paper. They first give an answer to Suzuki's question(2013), and then obtain several results of(**)-Haagerup property parallel to those of Haagerup property for C*-algebras. It is proved that a nuclear unital C*-algebra with a faithful tracial state always has the(**)-Haagerup property. Some heredity results concerning the(**)-Haagerup property are also proved.
文摘目的为了提升生成对抗网络汉字风格迁移的图像生成质量,实现汉字智能生成在字库产业中的实际应用,提出了一种基于直观汉字构形学的条件生成对抗网络字体生成优化方法(Optimizationof Conditional Fonts Generation with Chinese Character Configuration GANs,C^(3)-GAN)。方法建构了直观汉字构形模组(C^(3)Module),该模组包含了利于条件生成对抗网络进行汉字构形语义特征学习的全特征汉字字符集。C^(3)-GAN在条件生成对抗网络模型下进行字体生成训练,降低了必要训练样本数量,实现对字体生成效果的优化。结果使用C^(3)-GAN生成汉字图像的清晰度更高、字形更准确。在图像相似性定量评估中,使用C^(3)-GAN的实验组相比于其他模型,获得了更高的相似值和更小的误差值。结论使用C^(3)-GAN可以降低必要训练样本数量、提升汉字图像质量。在实际项目中具有一定的应用性和可操作性。
