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
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 characterise the positive cone of a real C^(*)-algebra geometrically.Given an open coneΩin a real Banach space V,with the closureΩ,we show thatΩis the interior of the positive cone of a unital real C^(*)-algebra...We characterise the positive cone of a real C^(*)-algebra geometrically.Given an open coneΩin a real Banach space V,with the closureΩ,we show thatΩis the interior of the positive cone of a unital real C^(*)-algebra if and only if it is a Finsler symmetric cone with an orientable extension,which is equivalent to the condition that V is,in an equivalent norm,the Hermitian part of a unital real C^(*)-algebra with the positive coneΩ.展开更多
We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an ap...We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an application,let A be an infinite dimensional simple unital C-algebra such that A has one of the above-listed properties.Suppose that α:G→Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property.Then the crossed product C^*-algebra C^*(G,A,α) also has the property under consideration.展开更多
In the current article,we prove the crossed product C^*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C^*-algebra is strongly quasidiagonal again.We also show that a just-infinite C^*-algebra...In the current article,we prove the crossed product C^*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C^*-algebra is strongly quasidiagonal again.We also show that a just-infinite C^*-algebra is quasidiagonal if and only if it is inner quasidiagonal.Finally,we compute the topological free entropy dimension in just-infinite C^*-algebras.展开更多
We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self abs...We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self absorbing C^(*)-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_(2)),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.展开更多
Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand s...Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^(*)-algebra C_(r)^(*)(■)of■is a unital simple approximately finite(AF)-dimensional C^(*)-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^(*)-algebra C_(r)^(*)(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^(*)-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_(r)^(*)(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^(*)-dynamical system(C_(r)^(*)(■),G).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
Elliott dimension drop interval algebra is an important class among all C^*-algebras in the classification theory.Especially,they are building stones of AHD algebra and the latter contains all AH algebras with the ide...Elliott dimension drop interval algebra is an important class among all C^*-algebras in the classification theory.Especially,they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth.In this paper,the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra.Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth.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
The paper is devoted to the study of generalized inductive limit of C*-algebras with coherent maps being completely positive contractions of order zero: the nuclear dimension of generalized inductive limit of C*-al...The paper is devoted to the study of generalized inductive limit of C*-algebras with coherent maps being completely positive contractions of order zero: the nuclear dimension of generalized inductive limit of C*-algebras with finite nuclear dimension is finite; the generalized inductive limits of C*-algebras with the α-comparison property also have the s-comparison property.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
This paper defines a pairing of two finite Hopf C^*-algebras A and B, and investigates the interactions between them. If the pairing is non-degenerate, then the quantum double construction is given. This construction...This paper defines a pairing of two finite Hopf C^*-algebras A and B, and investigates the interactions between them. If the pairing is non-degenerate, then the quantum double construction is given. This construction yields a new finite Hopf C^*-algebra D(A, B). The canonical embedding maps of A and B into the double are both isometric.展开更多
Let A be an infinite dimensional stably finite unital simple separable C^(*)-algebra.Let B■A be a stably(centrally)large subalgebra in A such that B is m-almost divisible(m-almost divisible,weakly(m,n)-divisible).The...Let A be an infinite dimensional stably finite unital simple separable C^(*)-algebra.Let B■A be a stably(centrally)large subalgebra in A such that B is m-almost divisible(m-almost divisible,weakly(m,n)-divisible).Then A is 2(m+1)-almost divisible(weakly m-almost divisible,secondly weakly(m,n)-divisible).展开更多
Let's take H as an infinite-dimensional Hilbert space and K(H) be the set of all compact operators on H. Using Spectral theorem for compact self-adjoint operators, we prove the Hyers-Ulam stability of Jensen z-deri...Let's take H as an infinite-dimensional Hilbert space and K(H) be the set of all compact operators on H. Using Spectral theorem for compact self-adjoint operators, we prove the Hyers-Ulam stability of Jensen z-derivations from K(H) into K(H).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 Engineering and Physical Sciences Research Council,UK(Grant No.EP/R044228/1).
文摘We characterise the positive cone of a real C^(*)-algebra geometrically.Given an open coneΩin a real Banach space V,with the closureΩ,we show thatΩis the interior of the positive cone of a unital real C^(*)-algebra if and only if it is a Finsler symmetric cone with an orientable extension,which is equivalent to the condition that V is,in an equivalent norm,the Hermitian part of a unital real C^(*)-algebra with the positive coneΩ.
基金Supported by National Natural Sciences Foundation of China(Grant Nos.11501357 and 11571008)。
文摘We show that the following properties of the C^*-algebras in a class Ω are inherited by simple unital C-algebras in the class TAΩ:(1)(m,n)-decomposable,(2) weakly(m,n)-divisible,(3) weak Riesz interpolation.As an application,let A be an infinite dimensional simple unital C-algebra such that A has one of the above-listed properties.Suppose that α:G→Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property.Then the crossed product C^*-algebra C^*(G,A,α) also has the property under consideration.
文摘In the current article,we prove the crossed product C^*-algebra by a Rokhlin action of finite group on a strongly quasidiagonal C^*-algebra is strongly quasidiagonal again.We also show that a just-infinite C^*-algebra is quasidiagonal if and only if it is inner quasidiagonal.Finally,we compute the topological free entropy dimension in just-infinite C^*-algebras.
基金supported by the Research Center for Operator Algebras at East China Normal University which is funded by the Science and Technology Commission of Shanghai Municipality (Grant No.13dz2260400)National Natural Science Foundation of China (Grant No.11531003)+1 种基金Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice (Grant No.1361431)the special fund for the Short-Term Training of Graduate Students from East China Normal University。
文摘We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C^(*)-algebra. We also show that if a unital C^(*)-algebra can be approximately embedded into some tensorially self absorbing C^(*)-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_(2)),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.
基金supported by National Natural Science Foundation of China(Grant Nos.11771379,11271224 and 11371290)。
文摘Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^(*)-algebra C_(r)^(*)(■)of■is a unital simple approximately finite(AF)-dimensional C^(*)-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^(*)-algebra C_(r)^(*)(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^(*)-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_(r)^(*)(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^(*)-dynamical system(C_(r)^(*)(■),G).
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Elliott dimension drop interval algebra is an important class among all C^*-algebras in the classification theory.Especially,they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth.In this paper,the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra.Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth.
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Supported by NSFC(Grant No.11371279)by the Fundamental Research Funds for the Central Universities
文摘The paper is devoted to the study of generalized inductive limit of C*-algebras with coherent maps being completely positive contractions of order zero: the nuclear dimension of generalized inductive limit of C*-algebras with finite nuclear dimension is finite; the generalized inductive limits of C*-algebras with the α-comparison property also have the s-comparison property.
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 National Natural Science Foundation of China(No.10301004)Excellent Young Scholars Research Fund of Beijing Institute of Technology(00Y07-25)
文摘This paper defines a pairing of two finite Hopf C^*-algebras A and B, and investigates the interactions between them. If the pairing is non-degenerate, then the quantum double construction is given. This construction yields a new finite Hopf C^*-algebra D(A, B). The canonical embedding maps of A and B into the double are both isometric.
基金supported by National Natural Sciences Foundation of China(11501357,11571008)supported by National Natural Sciences Foundation of China(11871375)。
文摘Let A be an infinite dimensional stably finite unital simple separable C^(*)-algebra.Let B■A be a stably(centrally)large subalgebra in A such that B is m-almost divisible(m-almost divisible,weakly(m,n)-divisible).Then A is 2(m+1)-almost divisible(weakly m-almost divisible,secondly weakly(m,n)-divisible).
文摘Let's take H as an infinite-dimensional Hilbert space and K(H) be the set of all compact operators on H. Using Spectral theorem for compact self-adjoint operators, we prove the Hyers-Ulam stability of Jensen z-derivations from K(H) into K(H).