Let A be a commutative unital C^(*)-algebra with the unit element e and M be a full Hilbert A-module.Denote by End_(A)(M)the algebra of all bounded A-linear mappings on M and by M′the set of all bounded A-linear mapp...Let A be a commutative unital C^(*)-algebra with the unit element e and M be a full Hilbert A-module.Denote by End_(A)(M)the algebra of all bounded A-linear mappings on M and by M′the set of all bounded A-linear mappings from M into A.In this paper,we prove that if there exists x_(0) in M and f_(0) in M′such that f_(0)(x_(0))=e,then every A-linear Lie triple derivation on End_(A)(M)is standard.展开更多
This paper,combined algebraical structure with analytical system,has studied the part of theory of C~*-modules over A by using the homolgical methods, where A is a commutative C~*-algebra over complex number field C. ...This paper,combined algebraical structure with analytical system,has studied the part of theory of C~*-modules over A by using the homolgical methods, where A is a commutative C~*-algebra over complex number field C. That is to say we have not only defined some relevant new concept,but also obtained some results about them.展开更多
The notion of weak Doi-Hopfπ-datum and weak Doi-Hopfπ-module are given as generalizations of an ordinary weak Doi-Hopf datum and weak Doi-Hopf module introduced in （Boehm, 2000）, also as a generalization of a Doi-...The notion of weak Doi-Hopfπ-datum and weak Doi-Hopfπ-module are given as generalizations of an ordinary weak Doi-Hopf datum and weak Doi-Hopf module introduced in （Boehm, 2000）, also as a generalization of a Doi-Hopfπ-module introduced in （Wang, 2004）. Then we also show that the functor forgetting action or coaction has an adjoint. Furthermore we explain how the notion of weak Doi-Hopfπ-datum is related to weak smash product. This paper presents our preliminary results on weak Doi-Hopf group modules.展开更多
It’s created a canonical Lie algebra in electrodynamics with all the “nice” algebraic and geometrical properties of an universal enveloping algebra with the goal of can to obtain generalizations in quantum electrod...It’s created a canonical Lie algebra in electrodynamics with all the “nice” algebraic and geometrical properties of an universal enveloping algebra with the goal of can to obtain generalizations in quantum electrodynamics theory of the TQFT, and the Universe based in lines and twistor bundles to the obtaining of irreducible unitary representations of the Lie groups SO(4) ?andO(3,1) , based in admissible representations of U(1) , and SU(n)? . The obtained object haves the advantages to be an algebraic or geometrical space at the same time. This same space of ￡-modules can explain and model different electromagnetic phenomena in superconductor and quantum processes where is necessary an organized transformation of the electromagnetic nature of the space- time and obtain nanotechnologies of the space-time and their elements.展开更多
A module is called a co-＊∞-module if it is co-selfsmall and ∞-quasi-injective. The properties and characterizations are investigated. When a module U is a co-＊∞-module, the functor Hom RU（-,U）is exact in Copre...A module is called a co-＊∞-module if it is co-selfsmall and ∞-quasi-injective. The properties and characterizations are investigated. When a module U is a co-＊∞-module, the functor Hom RU（-,U）is exact in Copres∞（U）. A module U is a co-＊∞-module if and only if U is co-selfsmall and for any exact sequence 0→M→UI→N→0 with M∈Copres∞（U） and I is a set, N∈Copres∞（U） is equivalent to Ext1R（N,U）→Ext1R（UI,U） is a monomorphism if and only if U is co-selfsmall and for any exact sequence 0→L→M→N→0 with L, N∈Copres∞（U）, N∈Copres∞（U） is equivalent to the induced sequence 0→Δ（N）→Δ（M）→Δ（L）→0 which is exact if and only if U induces a duality ΔUS：⊥USCopres∞（U）：ΔRU. Moreover, U is a co-＊n-module if and only if U is a co-＊∞-module and Copres∞（U）=Copresn（U）.展开更多
In this article, we introduce the notion of generalized derivations on Hilbert C＊-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen...In this article, we introduce the notion of generalized derivations on Hilbert C＊-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen type functional equationrf（x＋y/r）＋sg（x-y/s）=2h（x）for r, s ∈ R ／ {0} on Hilbert C＊-modules, where f, g, and h are mappings from a Hilbert C＊-module M to M.展开更多
In this paper we define and study chain conditions for Hilbert C＊-modules through their C＊-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that thes...In this paper we define and study chain conditions for Hilbert C＊-modules through their C＊-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C＊-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C＊-algebra with a faithful conditional expectation.展开更多
In this paper,we focus on combining the theories of fuzzy soft sets with Γ-modules,and establishing a new framework for fuzzy soft Γ-submodules.The main contributions of the paper are 3-fold.First,we present the con...In this paper,we focus on combining the theories of fuzzy soft sets with Γ-modules,and establishing a new framework for fuzzy soft Γ-submodules.The main contributions of the paper are 3-fold.First,we present the concepts of（R,S）-bi-Γ-submodules,quasi-Γ-submodules and regular Γ-modules.Meanwhile,some illustrative examples are given to show the rationality of the definitions introduced in this paper.Second,several new kinds of generalized fuzzy soft Γ-submodules are proposed,and related properties and mutual relationships are also investigated.Third,we discover some intrinsic connections between the generalized fuzzy soft Γ-submodules presented in this paper and crisp Γ-submodules,and describe the relationships between regular Γ-modules and the generalized fuzzy soft Γ-submodules presented in this paper.展开更多
In this note, we establish a new characterization on g-frames in Hilbert C;-modules from the operator-theoretic point of view, with which we provide a correction to one result recently obtained by Yao(Yao X Y. Some pr...In this note, we establish a new characterization on g-frames in Hilbert C;-modules from the operator-theoretic point of view, with which we provide a correction to one result recently obtained by Yao(Yao X Y. Some properties of g-frames in Hilbert C;-modules（in Chinese）. Acta Math. Sinica, 2011, 54（1）: 1–8.).展开更多
In this paper,we characterize a class of quasi-modular maps on Hilbert C~＊-modules which map a ＂rank one＂ adjointable operator to another rank one operator.
Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules ove...Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules over R are characterized by means of co-*-modules.展开更多
Compared with Zn^(2+),the current mainly reported charge carrier for zinc hybrid capacitors,small-hydrated-sized and light-weight NH_(4)^(+)is expected as a better one to mediate cathodic interfacial electrochemical b...Compared with Zn^(2+),the current mainly reported charge carrier for zinc hybrid capacitors,small-hydrated-sized and light-weight NH_(4)^(+)is expected as a better one to mediate cathodic interfacial electrochemical behaviors,yet has not been unraveled.Here we propose an NH_(4)^(+)-modulated cationic solvation strategy to optimize cathodic spatial charge distribution and achieve dynamic Zn^(2+)/NH_(4)^(+)co-storage for boosting Zinc hybrid capacitors.Owing to the hierarchical cationic solvated structure in hybrid Zn(CF_(3)SO_(3))_(2)–NH_4CF_(3)SO_(3)electrolyte,high-reactive Zn^(2+)and small-hydrate-sized NH_4(H_(2)O))(4)^(+)induce cathodic interfacial Helmholtz plane reconfiguration,thus effectively enhancing the spatial charge density to activate 20%capacity enhancement.Furthermore,cathodic interfacial adsorbed hydrated NH_(4)^(+)ions afford high-kinetics and ultrastable C···H(NH_(4)^(+))charge storage process due to a much lower desolvation energy barrier compared with heavy and rigid Zn(H_(2)O)_6^(2+)(5.81 vs.14.90 eV).Consequently,physical uptake and multielectron redox of Zn^(2+)/NH_(4)^(+)in carbon cathode enable the zinc capacitor to deliver high capacity(240 mAh g^(-1)at 0.5 A g^(-1)),large-current tolerance(130 mAh g^(-1)at 50 A g^(-1))and ultralong lifespan(400,000cycles).This study gives new insights into the design of cathode–electrolyte interfaces toward advanced zinc-based energy storage.展开更多
One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summa...One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.展开更多
Let M be a full Hilbert C＊-module over a C＊-algebra A, and let End^（.A4） be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End＊A（M） is an inner ...Let M be a full Hilbert C＊-module over a C＊-algebra A, and let End^（.A4） be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End＊A（M） is an inner derivation, and that if A is a-unital and commutative, then innerness of derivations on ＂compact＂ operators completely decides innerness of derivations on EndA（M）. If .4 is unital （no commutativity is assumed） such that every derivation of A is inner, then it is proved that every derivation of EndA（Ln（A）） is also inner, where Ln（A） denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist xo,yo ∈M such that 〈xo,yo〉 = 1, we characterize the linear A-module homomorphisms on EndA（M） which behave like derivations when acting on zero products.展开更多
The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z＋-modules over the near-group fusion ring K（Z3, 3） are explicitly classified. It turns...The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z＋-modules over the near-group fusion ring K（Z3, 3） are explicitly classified. It turns out that there are only four inequivalent irreducible Z＋-modules of rank 2 and two inequivalent irreducible Z＋-modules of rank 4 over K（Z3, 3）.展开更多
In this paper,we generalize the Takesaki-Takai duality theorem in Hilbert C~*-modules; that is to say,if (H,V,U) is a Kac-system,where H is a Hilbert space,V is a multiplicative unitary operator on H(?)H and U is a un...In this paper,we generalize the Takesaki-Takai duality theorem in Hilbert C~*-modules; that is to say,if (H,V,U) is a Kac-system,where H is a Hilbert space,V is a multiplicative unitary operator on H(?)H and U is a unitary operator on H,and if E is an (?)-compatible Hilbert (?)-module, then E×(?)×(?)K(H),where K(H) is the set of all compact operators on H,and (?) and (?) are Hopf C~*-algebras corresponding to the Kac-system (H,V,U).展开更多
In this paper, we obtain the boundedness of the fractional integral operators, the bilineax fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.
Theory of uncertainty reasoning base on l-module of true value field,in this paper,an extended l-module was proposed,some properties and lattice measure were discussed,then the lattice integral on l-module was gained.
In the paper, we introduce weak Bessel sequences and weak frames in a Hilbert C＊-module 74, and give a characterization of weak Bessel sequences, weak frames, normalized tight weak frames, and dual weak frames to eac...In the paper, we introduce weak Bessel sequences and weak frames in a Hilbert C＊-module 74, and give a characterization of weak Bessel sequences, weak frames, normalized tight weak frames, and dual weak frames to each other, respectively. Using .A-valued linear bounded operator U ： H → l^2（.A）, V＊U = I, a coustructing method of dual weak frame {xj^＊ ： j ∈ H} for a given weak frame {Xj ： j ∈ J} is obtained. Moreover, pseudo frame decompositions for 74 is given.展开更多
In this paper, first, we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Cheb...In this paper, first, we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev. Examples of those algebras are given including the algebras of continuous functions on compact sets. We also see some results in C＊-algebras and Hilbert C＊-modules. Next, by considering some conditions, we study Chebyshev of subalgebras in C＊-algebras.展开更多
基金Supported by the Shaanxi College Students Innovation and Entrepreneurship Training Program(Grant No.S202110708069)。
文摘Let A be a commutative unital C^(*)-algebra with the unit element e and M be a full Hilbert A-module.Denote by End_(A)(M)the algebra of all bounded A-linear mappings on M and by M′the set of all bounded A-linear mappings from M into A.In this paper,we prove that if there exists x_(0) in M and f_(0) in M′such that f_(0)(x_(0))=e,then every A-linear Lie triple derivation on End_(A)(M)is standard.
文摘This paper,combined algebraical structure with analytical system,has studied the part of theory of C~*-modules over A by using the homolgical methods, where A is a commutative C~*-algebra over complex number field C. That is to say we have not only defined some relevant new concept,but also obtained some results about them.
基金Project supported by the Program for New Century Excellent Talents in University (No. 04-0522), the National Science Foundation of Zhejiang Province of China (No. 102028), and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 704004)
文摘The notion of weak Doi-Hopfπ-datum and weak Doi-Hopfπ-module are given as generalizations of an ordinary weak Doi-Hopf datum and weak Doi-Hopf module introduced in （Boehm, 2000）, also as a generalization of a Doi-Hopfπ-module introduced in （Wang, 2004）. Then we also show that the functor forgetting action or coaction has an adjoint. Furthermore we explain how the notion of weak Doi-Hopfπ-datum is related to weak smash product. This paper presents our preliminary results on weak Doi-Hopf group modules.
文摘It’s created a canonical Lie algebra in electrodynamics with all the “nice” algebraic and geometrical properties of an universal enveloping algebra with the goal of can to obtain generalizations in quantum electrodynamics theory of the TQFT, and the Universe based in lines and twistor bundles to the obtaining of irreducible unitary representations of the Lie groups SO(4) ?andO(3,1) , based in admissible representations of U(1) , and SU(n)? . The obtained object haves the advantages to be an algebraic or geometrical space at the same time. This same space of ￡-modules can explain and model different electromagnetic phenomena in superconductor and quantum processes where is necessary an organized transformation of the electromagnetic nature of the space- time and obtain nanotechnologies of the space-time and their elements.
基金The National Natural Science Foundation of China (No.10971024)Specialized Research Fund for the Doctoral Program of Higher Education (No.200802860024)
文摘A module is called a co-＊∞-module if it is co-selfsmall and ∞-quasi-injective. The properties and characterizations are investigated. When a module U is a co-＊∞-module, the functor Hom RU（-,U）is exact in Copres∞（U）. A module U is a co-＊∞-module if and only if U is co-selfsmall and for any exact sequence 0→M→UI→N→0 with M∈Copres∞（U） and I is a set, N∈Copres∞（U） is equivalent to Ext1R（N,U）→Ext1R（UI,U） is a monomorphism if and only if U is co-selfsmall and for any exact sequence 0→L→M→N→0 with L, N∈Copres∞（U）, N∈Copres∞（U） is equivalent to the induced sequence 0→Δ（N）→Δ（M）→Δ（L）→0 which is exact if and only if U induces a duality ΔUS：⊥USCopres∞（U）：ΔRU. Moreover, U is a co-＊n-module if and only if U is a co-＊∞-module and Copres∞（U）=Copresn（U）.
文摘In this article, we introduce the notion of generalized derivations on Hilbert C＊-modules. We use a fixed-point method to prove the generalized Hyers-Ulam-Rassias stability associated to the Pexiderized Cauchy-Jensen type functional equationrf（x＋y/r）＋sg（x-y/s）=2h（x）for r, s ∈ R ／ {0} on Hilbert C＊-modules, where f, g, and h are mappings from a Hilbert C＊-module M to M.
文摘In this paper we define and study chain conditions for Hilbert C＊-modules through their C＊-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C＊-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C＊-algebra with a faithful conditional expectation.
基金Supported by the National Natural Science Foundation of China (61175055)the Innovation Term of Higher Education of Hubei Province,China (T201109)+1 种基金the Natural Science Foundation of Hubei Province (2012FFB01101)the Natural Science Foundation of Education Committee of Hubei Province (D20131903)
文摘In this paper,we focus on combining the theories of fuzzy soft sets with Γ-modules,and establishing a new framework for fuzzy soft Γ-submodules.The main contributions of the paper are 3-fold.First,we present the concepts of（R,S）-bi-Γ-submodules,quasi-Γ-submodules and regular Γ-modules.Meanwhile,some illustrative examples are given to show the rationality of the definitions introduced in this paper.Second,several new kinds of generalized fuzzy soft Γ-submodules are proposed,and related properties and mutual relationships are also investigated.Third,we discover some intrinsic connections between the generalized fuzzy soft Γ-submodules presented in this paper and crisp Γ-submodules,and describe the relationships between regular Γ-modules and the generalized fuzzy soft Γ-submodules presented in this paper.
基金The NSF(11271148,11561057)of Chinathe NSF(20151BAB201007)of Jiangxi Provincethe Science and Technology Project(GJJ151061)of Jiangxi Education Department
文摘In this note, we establish a new characterization on g-frames in Hilbert C;-modules from the operator-theoretic point of view, with which we provide a correction to one result recently obtained by Yao(Yao X Y. Some properties of g-frames in Hilbert C;-modules（in Chinese）. Acta Math. Sinica, 2011, 54（1）: 1–8.).
基金Supported by the National Natural Science Foundation of China (Grant No. 11171151)
文摘In this paper,we characterize a class of quasi-modular maps on Hilbert C~＊-modules which map a ＂rank one＂ adjointable operator to another rank one operator.
文摘Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules over R are characterized by means of co-*-modules.
基金financially supported by the National Natural Science Foundation of China(Nos.22272118,22172111 and 22309134)the Science and Technology Commission of Shanghai Municipality,China(Nos.22ZR1464100,20ZR1460300 and 19DZ2271500)+3 种基金China Postdoctoral Science Foundation(2022M712402)Shanghai Rising-Star Program(23YF1449200)Zhejiang Provincial Science and Technology Project(2022C01182)the Fundamental Research Funds for the Central Universities(22120210529 and 2023-3-YB-07)。
文摘Compared with Zn^(2+),the current mainly reported charge carrier for zinc hybrid capacitors,small-hydrated-sized and light-weight NH_(4)^(+)is expected as a better one to mediate cathodic interfacial electrochemical behaviors,yet has not been unraveled.Here we propose an NH_(4)^(+)-modulated cationic solvation strategy to optimize cathodic spatial charge distribution and achieve dynamic Zn^(2+)/NH_(4)^(+)co-storage for boosting Zinc hybrid capacitors.Owing to the hierarchical cationic solvated structure in hybrid Zn(CF_(3)SO_(3))_(2)–NH_4CF_(3)SO_(3)electrolyte,high-reactive Zn^(2+)and small-hydrate-sized NH_4(H_(2)O))(4)^(+)induce cathodic interfacial Helmholtz plane reconfiguration,thus effectively enhancing the spatial charge density to activate 20%capacity enhancement.Furthermore,cathodic interfacial adsorbed hydrated NH_(4)^(+)ions afford high-kinetics and ultrastable C···H(NH_(4)^(+))charge storage process due to a much lower desolvation energy barrier compared with heavy and rigid Zn(H_(2)O)_6^(2+)(5.81 vs.14.90 eV).Consequently,physical uptake and multielectron redox of Zn^(2+)/NH_(4)^(+)in carbon cathode enable the zinc capacitor to deliver high capacity(240 mAh g^(-1)at 0.5 A g^(-1)),large-current tolerance(130 mAh g^(-1)at 50 A g^(-1))and ultralong lifespan(400,000cycles).This study gives new insights into the design of cathode–electrolyte interfaces toward advanced zinc-based energy storage.
文摘One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.
基金supported by National Natural Science Foundation of China(Grant No.11171151)Natural Science Foundation of Jiangsu Province of China(Grant No.BK2011720)supported by Singapore Ministry of Education Academic Research Fund Tier1(Grant No.R-146-000-136-112)
文摘Let M be a full Hilbert C＊-module over a C＊-algebra A, and let End^（.A4） be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End＊A（M） is an inner derivation, and that if A is a-unital and commutative, then innerness of derivations on ＂compact＂ operators completely decides innerness of derivations on EndA（M）. If .4 is unital （no commutativity is assumed） such that every derivation of A is inner, then it is proved that every derivation of EndA（Ln（A）） is also inner, where Ln（A） denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist xo,yo ∈M such that 〈xo,yo〉 = 1, we characterize the linear A-module homomorphisms on EndA（M） which behave like derivations when acting on zero products.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No. 11471282）.
文摘The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z＋-modules over the near-group fusion ring K（Z3, 3） are explicitly classified. It turns out that there are only four inequivalent irreducible Z＋-modules of rank 2 and two inequivalent irreducible Z＋-modules of rank 4 over K（Z3, 3）.
基金Supported by NSF 10301004,NSF 10171098Yantai University PHD Foundation SX03B14
文摘In this paper,we generalize the Takesaki-Takai duality theorem in Hilbert C~*-modules; that is to say,if (H,V,U) is a Kac-system,where H is a Hilbert space,V is a multiplicative unitary operator on H(?)H and U is a unitary operator on H,and if E is an (?)-compatible Hilbert (?)-module, then E×(?)×(?)K(H),where K(H) is the set of all compact operators on H,and (?) and (?) are Hopf C~*-algebras corresponding to the Kac-system (H,V,U).
基金Supported by the National Natural Science Foundation of China(11271330)
文摘In this paper, we obtain the boundedness of the fractional integral operators, the bilineax fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.
基金Supported by NSF of the Education Department of Henan Province(2009A110017)
文摘Theory of uncertainty reasoning base on l-module of true value field,in this paper,an extended l-module was proposed,some properties and lattice measure were discussed,then the lattice integral on l-module was gained.
基金Supported by the Emphasis Supported Subject Foundation of Shanxi Province（20055026） Supported by the Emphasis Science Foundation of Yuncheng University（20060103）
文摘In the paper, we introduce weak Bessel sequences and weak frames in a Hilbert C＊-module 74, and give a characterization of weak Bessel sequences, weak frames, normalized tight weak frames, and dual weak frames to each other, respectively. Using .A-valued linear bounded operator U ： H → l^2（.A）, V＊U = I, a coustructing method of dual weak frame {xj^＊ ： j ∈ H} for a given weak frame {Xj ： j ∈ J} is obtained. Moreover, pseudo frame decompositions for 74 is given.
文摘In this paper, first, we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev. Examples of those algebras are given including the algebras of continuous functions on compact sets. We also see some results in C＊-algebras and Hilbert C＊-modules. Next, by considering some conditions, we study Chebyshev of subalgebras in C＊-algebras.
