Using fixed point methods, we prove the Hyers–Ulam–Rassias stability and superstability of Jordan homomorphisms （Jordan ＊-homomorphisms）, and Jordan derivations （Jordan ＊-derivations） on Banach algebras （C＊-...Using fixed point methods, we prove the Hyers–Ulam–Rassias stability and superstability of Jordan homomorphisms （Jordan ＊-homomorphisms）, and Jordan derivations （Jordan ＊-derivations） on Banach algebras （C＊-algebras） for the generalized Jensen–type functional equationwhere r is a fixed positive real number in （1, ∞）.展开更多
Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the sta...Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.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
We show that every unital invertibility preserving linear map from a von Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism;this gives an affirmative answer to a problem of Kaplansky for all vo...We show that every unital invertibility preserving linear map from a von Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism;this gives an affirmative answer to a problem of Kaplansky for all von Neumann algebras.For a unital linear map Φ from a semi-simple complex Banach algebra onto another,we also show that the following statements are equivalent:(1) Φ is an homomorphism;(2)Φ is completely invertibility preserving;(3)Φ is 2-invertibility preserving.展开更多
Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by Mn（R） the ring of all n x n matrices over R. Let （Jn（R）） be the additive subgroup of Mn（...Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by Mn（R） the ring of all n x n matrices over R. Let （Jn（R）） be the additive subgroup of Mn（R） generated additively by all idempotent matrices. Let JJ = （Jn（R）） or Mn（R）. We describe the additive preservers of idempotence from JJ to Mm（R） when 2 is a unit of R. Thereby, we also characterize the Jordan （respectively, ring and ring anti-） homomorphisms from Mn （R） to Mm （R） when 2 is a unit of R.展开更多
We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability th...We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam＇s problem.展开更多
文摘Using fixed point methods, we prove the Hyers–Ulam–Rassias stability and superstability of Jordan homomorphisms （Jordan ＊-homomorphisms）, and Jordan derivations （Jordan ＊-derivations） on Banach algebras （C＊-algebras） for the generalized Jensen–type functional equationwhere r is a fixed positive real number in （1, ∞）.
文摘Let R be a prime ring, L a non-central Lie ideal of R and g a non-zero generalized derivation of R. If g acts as a Jordan homomorphism on L, then either g（x） = x for all x ∈ R, or char（R） = 2, R satisfies the standard identity s4（x1, x2, x3, x4）, L is commutative and u2 ∈ Z（R）, for any u C L. We also examine some consequences of this result related to generalized derivations which act as Jordan homomorphisms on the set [I, I], where I is a non-zero right ideal of R.
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.
基金supported by NNSFC (10071046)PNSFS (981009)+1 种基金PYSFS(20031009)China Postdoctoral Science Foundation
文摘We show that every unital invertibility preserving linear map from a von Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism;this gives an affirmative answer to a problem of Kaplansky for all von Neumann algebras.For a unital linear map Φ from a semi-simple complex Banach algebra onto another,we also show that the following statements are equivalent:(1) Φ is an homomorphism;(2)Φ is completely invertibility preserving;(3)Φ is 2-invertibility preserving.
基金This work is supported in part by the Chinese Natural Science Foundation under Grant No. 10671026 and the Postdoctoral Fund of Heilongjiang Province
文摘Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by Mn（R） the ring of all n x n matrices over R. Let （Jn（R）） be the additive subgroup of Mn（R） generated additively by all idempotent matrices. Let JJ = （Jn（R）） or Mn（R）. We describe the additive preservers of idempotence from JJ to Mm（R） when 2 is a unit of R. Thereby, we also characterize the Jordan （respectively, ring and ring anti-） homomorphisms from Mn （R） to Mm （R） when 2 is a unit of R.
基金partly supported by the National Natural Science Foundation of China(19771056)
文摘We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam＇s problem.
