In this paper,we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen's equations in Banach modules over a unital C~*-algebra.It is applied to show the stability of universal Jensen's equatio...In this paper,we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen's equations in Banach modules over a unital C~*-algebra.It is applied to show the stability of universal Jensen's equations in a Hilbert module over a unital C~*-algebra.Moreover,we prove the stability of linear operators in a Hilbert module over a unitat C~*-algebra.展开更多
In this paper,we study real Banach * algebras systematically.We present the right form of Pták’s inequality[1,4]in the real case,and generalize the results of Vukman in[3]to the general case(algebras with or wit...In this paper,we study real Banach * algebras systematically.We present the right form of Pták’s inequality[1,4]in the real case,and generalize the results of Vukman in[3]to the general case(algebras with or without an identity).Moreover,this paper is a real analogue of Pták’s work[1] in the complex case.展开更多
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 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
Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irration...Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr（Aω） = tr（Aρ）. It is proved that the set of all C*-algebras of sections of locally trivial C*-algebra bundles over S2 with fibres Aω. has a group structure, denoted by π1（Aut（Aω.））, which is isomorphic to Z if d > 1 and {0} if d > 1. Let Bcd be a cd-homogeneous C*-algebra over S2 x T2 of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torns Sρcd is defined by twisting C*（T2 x Zm-2） in Bcd C* （Z（m-2）） by a totally skew multiplier ρ on T2 x Z（m-2）. It is shown that Sρcd Mp∞ is isomorphic to C（S2） C* （T2 x Zm-2, ρ） Mcd（C） Mp∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.展开更多
We introduce a special tracial Rokhlin property for unital C~＊-algebras. Let A be a unital tracial rank zero C~＊-algebra（or tracial rank no more than one C~＊-algebra）. Suppose that α ： G → Aut（A） is an actio...We introduce a special tracial Rokhlin property for unital C~＊-algebras. Let A be a unital tracial rank zero C~＊-algebra（or tracial rank no more than one C~＊-algebra）. Suppose that α ： G → Aut（A） is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~＊-algebra. Then, the crossed product C~＊-algebra C~＊（G, A, α） has tracia rank zero（or has tracial rank no more than one）. In fact,we get a more general results.展开更多
Let A be a finite-dimensional algebra over the real number field.We prove that the repetitive algebra A admits the dichotomy property of representation type,i.e.,A is either of discrete representation type or of stron...Let A be a finite-dimensional algebra over the real number field.We prove that the repetitive algebra A admits the dichotomy property of representation type,i.e.,A is either of discrete representation type or of strongly unbounded type.展开更多
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
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 study diagonal invariant ideals of topologically graded C*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the...We study diagonal invariant ideals of topologically graded C*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author's previous works on this topic.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
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
Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n = 0. An additive mapping δ from R into M is called an(m, n)-Jordan derivation if(m +n)δ(A^2) = 2 mAδ(A) + 2nδ(A)A for every ...Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n = 0. An additive mapping δ from R into M is called an(m, n)-Jordan derivation if(m +n)δ(A^2) = 2 mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every(m, n)-Jordan derivation with m = n from a C*-algebra into its Banach bimodule is zero. An additive mappingδ from R into M is called a(m, n)-Jordan derivable mapping at W in R if(m + n)δ(AB + BA) =2mδ(A)B + 2 mδ(B)A + 2 nAδ(B) + 2 nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left(right) separating set generated algebraically by all idempotents in A, then every(m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital(A, B)-bimodule and U = [A M N B] is a generalized matrix algebra, then every(m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.展开更多
The design of this paper is to present the first installment of a complete and final theory of rational human intelligence. The theory is mathematical in the strictest possible sense. The mathematics involved is stric...The design of this paper is to present the first installment of a complete and final theory of rational human intelligence. The theory is mathematical in the strictest possible sense. The mathematics involved is strictly digital—not quantitative in the manner that what is usually thought of as mathematics is quantitative. It is anticipated at this time that the exclusively digital nature of rational human intelligence exhibits four flavors of digitality, apparently no more, and that each flavor will require a lengthy study in its own right. (For more information,please refer to the PDF.)展开更多
基金supported by Korea Research Foundation Grant KRF-2002-041-C00014
文摘In this paper,we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen's equations in Banach modules over a unital C~*-algebra.It is applied to show the stability of universal Jensen's equations in a Hilbert module over a unital C~*-algebra.Moreover,we prove the stability of linear operators in a Hilbert module over a unitat C~*-algebra.
基金This work is supported in part by research grants of NSF of China (the first author)of the Chinese University of Hong Kong (both authors).
文摘In this paper,we study real Banach * algebras systematically.We present the right form of Pták’s inequality[1,4]in the real case,and generalize the results of Vukman in[3]to the general case(algebras with or without an identity).Moreover,this paper is a real analogue of Pták’s work[1] in the complex case.
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）.
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.
基金Project supported by the grant No. 1999-2-102-001-3 from the Interdisciplinary Research Program Year of the KOSEF
文摘Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aω of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr（Aω） = tr（Aρ）. It is proved that the set of all C*-algebras of sections of locally trivial C*-algebra bundles over S2 with fibres Aω. has a group structure, denoted by π1（Aut（Aω.））, which is isomorphic to Z if d > 1 and {0} if d > 1. Let Bcd be a cd-homogeneous C*-algebra over S2 x T2 of which no non-trivial matrix algebra can be factored out. The spherical noncommutative torns Sρcd is defined by twisting C*（T2 x Zm-2） in Bcd C* （Z（m-2）） by a totally skew multiplier ρ on T2 x Z（m-2）. It is shown that Sρcd Mp∞ is isomorphic to C（S2） C* （T2 x Zm-2, ρ） Mcd（C） Mp∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.
文摘We introduce a special tracial Rokhlin property for unital C~＊-algebras. Let A be a unital tracial rank zero C~＊-algebra（or tracial rank no more than one C~＊-algebra）. Suppose that α ： G → Aut（A） is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~＊-algebra. Then, the crossed product C~＊-algebra C~＊（G, A, α） has tracia rank zero（or has tracial rank no more than one）. In fact,we get a more general results.
基金supported by the National Natural Science Foundation of China(Grant No.11961007)Science Technology Foundation of Guizhou Province(Grant Nos.[2018]1021,[2020]1Y405).
文摘Let A be a finite-dimensional algebra over the real number field.We prove that the repetitive algebra A admits the dichotomy property of representation type,i.e.,A is either of discrete representation type or of strongly unbounded type.
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.
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 National Natural Science Foundation of China(10371051)
文摘We study diagonal invariant ideals of topologically graded C*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author's previous works on this topic.
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).
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(Grant Nos.11801342 and 11801005)
文摘Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n = 0. An additive mapping δ from R into M is called an(m, n)-Jordan derivation if(m +n)δ(A^2) = 2 mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every(m, n)-Jordan derivation with m = n from a C*-algebra into its Banach bimodule is zero. An additive mappingδ from R into M is called a(m, n)-Jordan derivable mapping at W in R if(m + n)δ(AB + BA) =2mδ(A)B + 2 mδ(B)A + 2 nAδ(B) + 2 nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left(right) separating set generated algebraically by all idempotents in A, then every(m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital(A, B)-bimodule and U = [A M N B] is a generalized matrix algebra, then every(m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.
文摘The design of this paper is to present the first installment of a complete and final theory of rational human intelligence. The theory is mathematical in the strictest possible sense. The mathematics involved is strictly digital—not quantitative in the manner that what is usually thought of as mathematics is quantitative. It is anticipated at this time that the exclusively digital nature of rational human intelligence exhibits four flavors of digitality, apparently no more, and that each flavor will require a lengthy study in its own right. (For more information,please refer to the PDF.)
