In this article, we prove the Hyers-Ulam-Rassias stability of the following Cauchy-Jensen functional inequality:‖f (x) + f (y) + 2f (z) + 2f (w)‖ ≤‖ 2f x + y2 + z + w ‖(0.1)This is applied to inv...In this article, we prove the Hyers-Ulam-Rassias stability of the following Cauchy-Jensen functional inequality:‖f (x) + f (y) + 2f (z) + 2f (w)‖ ≤‖ 2f x + y2 + z + w ‖(0.1)This is applied to investigate isomorphisms between C*-algebras, Lie C*-algebras and JC*-algebras, and derivations on C*-algebras, Lie C*-algebras and JC*-algebras, associated with the Cauchy-Jensen functional equation 2f (x + y/2 + z + w) = f(x) + f(y) + 2f(z) + 2f(w).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 define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better...In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.展开更多
Perturbation problem of operator algebras was first introduced by Kadison and Kastler. In this short note, we consider the uniform perturbation of two classes of operator algebras, i.e., MF algebras and quasidiagonal ...Perturbation problem of operator algebras was first introduced by Kadison and Kastler. In this short note, we consider the uniform perturbation of two classes of operator algebras, i.e., MF algebras and quasidiagonal C*-algebras. We show that the sets of MF algebras and quasidiagonal C*-algebras of a given C*-algebra are closed under the perturbation of uniform norm.展开更多
The authors show that ifΘ=(θ_(jk))is a 3×3 totally irrational real skewsymmetric matrix,whereθ_(jk)∈[0,1)for j,k=1,2,3,then for anyε>0,there existsδ>0 satisfying the following:For any unital C^(*)-alg...The authors show that ifΘ=(θ_(jk))is a 3×3 totally irrational real skewsymmetric matrix,whereθ_(jk)∈[0,1)for j,k=1,2,3,then for anyε>0,there existsδ>0 satisfying the following:For any unital C^(*)-algebra A with the cancellation property,strict comparison and nonempty tracial state space,any four unitaries u1,u2,u3,w∈A such that(1)■,wujw-1=uj-1,w2=1A for j,k=1,2,3;(2)τ(aw)=0 and■for all n∈N,all a∈C^(*)(u1,u2,u3),j,k=1,2,3 and all tracial statesτon A,where C^(*)(u1,u2,u3)is the C^(*)-subalgebra generated by u1,u2 and u3,there exists a 4-tuple of unitaries■in A such that■and■for j,k=1,2,3.The above conclusion is also called that the rotation relations of three unitaries with the flip action is stable under the above conditions.展开更多
We show that the following properties of the C*-algebras in a class P are inherited by simple unital C*-algebras in the class of asymptotically tracially in P :(1) n-comparison,(2) α-comparison(1 ≤ α < ∞).
Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).
All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as ...All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than 1.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
Let ,4 and B be unital C*-algebras, and let J∈A,L∈B be Hermitian invertible elements. For every T∈A and S∈B, define T^+J=J^-1T*J and ,S^+L=^L-1,S*L. Then in such a way we endow the C*-algebras A and B with i...Let ,4 and B be unital C*-algebras, and let J∈A,L∈B be Hermitian invertible elements. For every T∈A and S∈B, define T^+J=J^-1T*J and ,S^+L=^L-1,S*L. Then in such a way we endow the C*-algebras A and B with indefinite structures. We characterize firstly the Jordan (J, L)+homomorphisms on C*-algebras. As applications, we further classify the bounded linear maps Ф: A →B preserving (J, L)-unitary elements. When ,4 = B(H) and B=B(K), where H and K are infinite dimensional and complete indefinite inner product spaces on real or complex fields, we prove that indefinite-unitary preserving bounded linear surjections are of the form T→UVTV^-1 (任意T∈B(H)) or T→UVT^+V^-1(任意T∈B(H)), where U∈B(K) is indefinite unitary and, V : H→ K is generalized indefinite unitary in the first form and generalized indefinite anti-unitary in the second one. Some results on indefinite orthogonality preserving additive maps are also given.展开更多
The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixti...The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixtieth anniversary of the problem, we revisit the problem in its original formulation and also explore its transition to a result with wide ranging applications. We also describe how the notion of “sparsification” transcended various fields and how this notion led to resolution of the problem.展开更多
In this paper, we consider the norms related to spectral geometric means and geometric means. When A and B are positive and invertible, we have ||A<sup>-1</sup>#B|| ≤ ||A<sup>-1</sup>σ<sub...In this paper, we consider the norms related to spectral geometric means and geometric means. When A and B are positive and invertible, we have ||A<sup>-1</sup>#B|| ≤ ||A<sup>-1</sup>σ<sub>s</sub>B||. Let H be a Hilbert space and B(H) be the set of all bounded linear operators on H. Let A ∈ B(H). If ||A#X|| = ||Aσ<sub>s</sub>X||, ?X ∈ B(H)<sup>++</sup>, then A is a scalar. When is a C*-algebra and for any , we have that ||logA#B|| = ||logAσ<sub>s</sub>B||, then is commutative.展开更多
It is shown that every almost *-homomorphism h : A→B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x∈A, and that every almost linear mapping h...It is shown that every almost *-homomorphism h : A→B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x∈A, and that every almost linear mapping h : A→B is a *-homomorphism when h(2^nu o y) - h(2^nu) o h(y), h(3^nu o y) - h(3^nu) o h(y) or h(q^nu o y) = h(q^nu) o h(y) for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost *-homomorphism h : A→B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x ∈A.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
In this paper, it is shown that the regular representation and regular covariant representation of the crossed products A×α G correspond to the twisted multiplicative unitary operators, where A is a Woronowicz C...In this paper, it is shown that the regular representation and regular covariant representation of the crossed products A×α G correspond to the twisted multiplicative unitary operators, where A is a Woronowicz C*-algebra acted upon by a discrete group G. Meanwhile, it is also shown that the regular covariant C*-algebra is the Woronowicz C*-algebra which corresponds to a multiplicative unitary. Finally, an explicit description of the multiplicative unitary operator for C(SUq(2))×α Z is given in terms of those of the Woronowicz C*-algebra C(SUq(2)) and the discrete group G.展开更多
We construct a class of C*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ,the closure of the seminorm ||[Ml1,·]|| on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C*-al...We construct a class of C*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ,the closure of the seminorm ||[Ml1,·]|| on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C*-algebra C*r(Γ, σ) for the pointwise multiplication operator Mlon l2(Γ), induced by a proper length function l on Γ with the property of bounded θ-dilation. Moreover, the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.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
Let A be a separable simple C*-algebra. For each a(≠0) in A, there exists a separable faithful and irreducible * representation (π, Hπ) on A such that π(a) has a non-trivial invariant subspace in Hπ.
We show that the following classes of C*-algebras in the classes t are inherited by simple unital C*-algebras in the classes TAft : (1) simple unital purely infinite C*-algebras, (2) unital isometrically rich ...We show that the following classes of C*-algebras in the classes t are inherited by simple unital C*-algebras in the classes TAft : (1) simple unital purely infinite C*-algebras, (2) unital isometrically rich C*-algebras, (3) unital Riesz interpolation C*-algebras.展开更多
We study K-theory of continuous deformations of C*-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz alge...We study K-theory of continuous deformations of C*-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.展开更多
基金supported by the Daejin University grants in 2010
文摘In this article, we prove the Hyers-Ulam-Rassias stability of the following Cauchy-Jensen functional inequality:‖f (x) + f (y) + 2f (z) + 2f (w)‖ ≤‖ 2f x + y2 + z + w ‖(0.1)This is applied to investigate isomorphisms between C*-algebras, Lie C*-algebras and JC*-algebras, and derivations on C*-algebras, Lie C*-algebras and JC*-algebras, associated with the Cauchy-Jensen functional equation 2f (x + y/2 + z + w) = f(x) + f(y) + 2f(z) + 2f(w).
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partially supported by the Natural Sciences and Engineering Research Council of Canada(2019-03907)。
文摘In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.
文摘Perturbation problem of operator algebras was first introduced by Kadison and Kastler. In this short note, we consider the uniform perturbation of two classes of operator algebras, i.e., MF algebras and quasidiagonal C*-algebras. We show that the sets of MF algebras and quasidiagonal C*-algebras of a given C*-algebra are closed under the perturbation of uniform norm.
基金supported by the National Natural Science Foundation of China(Nos.11401256,11801219,11501249,11871342)the Scientific Research Fund of Zhejiang Provincial Education Department(No.Y202249575)the Zhejiang Provincial Natural Science Foundation of China(No.LQ13A010016)。
文摘The authors show that ifΘ=(θ_(jk))is a 3×3 totally irrational real skewsymmetric matrix,whereθ_(jk)∈[0,1)for j,k=1,2,3,then for anyε>0,there existsδ>0 satisfying the following:For any unital C^(*)-algebra A with the cancellation property,strict comparison and nonempty tracial state space,any four unitaries u1,u2,u3,w∈A such that(1)■,wujw-1=uj-1,w2=1A for j,k=1,2,3;(2)τ(aw)=0 and■for all n∈N,all a∈C^(*)(u1,u2,u3),j,k=1,2,3 and all tracial statesτon A,where C^(*)(u1,u2,u3)is the C^(*)-subalgebra generated by u1,u2 and u3,there exists a 4-tuple of unitaries■in A such that■and■for j,k=1,2,3.The above conclusion is also called that the rotation relations of three unitaries with the flip action is stable under the above conditions.
基金Supported by the National Natural Sciences Foundation of China (Grant No. 11871375)。
文摘We show that the following properties of the C*-algebras in a class P are inherited by simple unital C*-algebras in the class of asymptotically tracially in P :(1) n-comparison,(2) α-comparison(1 ≤ α < ∞).
文摘Let A be an unital C*-algebra, a, x and y are elements in A. In this paper, we present a method how to calculate the Moore-Penrose inverse of a- xy*and investigate the expression for some new special cases of(a- xy*).
基金The author was supported by grant No. 1999-2-102-001-3 from the interdis- ciplinary research program year of the KOSEF.
文摘All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than 1.
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.
基金The NNSF (10471082) of Chinathe YSF (20031009) of Shanxi ProvinceTsinghua Basic Research Foundation
文摘Let ,4 and B be unital C*-algebras, and let J∈A,L∈B be Hermitian invertible elements. For every T∈A and S∈B, define T^+J=J^-1T*J and ,S^+L=^L-1,S*L. Then in such a way we endow the C*-algebras A and B with indefinite structures. We characterize firstly the Jordan (J, L)+homomorphisms on C*-algebras. As applications, we further classify the bounded linear maps Ф: A →B preserving (J, L)-unitary elements. When ,4 = B(H) and B=B(K), where H and K are infinite dimensional and complete indefinite inner product spaces on real or complex fields, we prove that indefinite-unitary preserving bounded linear surjections are of the form T→UVTV^-1 (任意T∈B(H)) or T→UVT^+V^-1(任意T∈B(H)), where U∈B(K) is indefinite unitary and, V : H→ K is generalized indefinite unitary in the first form and generalized indefinite anti-unitary in the second one. Some results on indefinite orthogonality preserving additive maps are also given.
文摘The Kadison-Singer problem has variants in different branches of the sciences and one of these variants was proved in 2013. Based on the idea of “sparsification” and with its origins in quantum physics, at the sixtieth anniversary of the problem, we revisit the problem in its original formulation and also explore its transition to a result with wide ranging applications. We also describe how the notion of “sparsification” transcended various fields and how this notion led to resolution of the problem.
文摘In this paper, we consider the norms related to spectral geometric means and geometric means. When A and B are positive and invertible, we have ||A<sup>-1</sup>#B|| ≤ ||A<sup>-1</sup>σ<sub>s</sub>B||. Let H be a Hilbert space and B(H) be the set of all bounded linear operators on H. Let A ∈ B(H). If ||A#X|| = ||Aσ<sub>s</sub>X||, ?X ∈ B(H)<sup>++</sup>, then A is a scalar. When is a C*-algebra and for any , we have that ||logA#B|| = ||logAσ<sub>s</sub>B||, then is commutative.
基金Grant No.R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.NNSF of China and NSF of Shanxi Province
文摘It is shown that every almost *-homomorphism h : A→B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x∈A, and that every almost linear mapping h : A→B is a *-homomorphism when h(2^nu o y) - h(2^nu) o h(y), h(3^nu o y) - h(3^nu) o h(y) or h(q^nu o y) = h(q^nu) o h(y) for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost *-homomorphism h : A→B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r 〉 1) for all x ∈A.
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supported by the National Natural Science Foundation of China(Grant Nos.10301004&10171098)Yantai University PhD Foundation(Grant No.SX03B14).
文摘In this paper, it is shown that the regular representation and regular covariant representation of the crossed products A×α G correspond to the twisted multiplicative unitary operators, where A is a Woronowicz C*-algebra acted upon by a discrete group G. Meanwhile, it is also shown that the regular covariant C*-algebra is the Woronowicz C*-algebra which corresponds to a multiplicative unitary. Finally, an explicit description of the multiplicative unitary operator for C(SUq(2))×α Z is given in terms of those of the Woronowicz C*-algebra C(SUq(2)) and the discrete group G.
基金supported by National Natural Science Foundation of China(Grant Nos.11171109 and 11801177)the Science and Technology Commission of Shanghai Municipality(Grant No.18dz2271000)。
文摘We construct a class of C*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ,the closure of the seminorm ||[Ml1,·]|| on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C*-algebra C*r(Γ, σ) for the pointwise multiplication operator Mlon l2(Γ), induced by a proper length function l on Γ with the property of bounded θ-dilation. Moreover, the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.
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This work was supported by the National Natural Science Foundation of China (Grant No. 10101026).
文摘Let A be a separable simple C*-algebra. For each a(≠0) in A, there exists a separable faithful and irreducible * representation (π, Hπ) on A such that π(a) has a non-trivial invariant subspace in Hπ.
文摘We show that the following classes of C*-algebras in the classes t are inherited by simple unital C*-algebras in the classes TAft : (1) simple unital purely infinite C*-algebras, (2) unital isometrically rich C*-algebras, (3) unital Riesz interpolation C*-algebras.
文摘We study K-theory of continuous deformations of C*-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.