Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I....Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I.For all A,B∈M,define by A■B=AB+BA the Jordan product of A and B,A·_(η)B=AB+ηBA^(*)the Jordan η-*-product of A and B,respectively.Let Φ and Φ^(-1)preserve the mixed Jordan triple η-*-products.It is proved that Φ is a linear *-isomorphism if η is not real and Φ is the sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η is real.展开更多
本文研究了在von Neumann代数框架下,投影运算后等价的保持情形,特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设E、F、N为von Neumann代数ℳ中的三个有限投影,且E~F。当NE=NF=0时,那么有N+E~N+F,N∧E~N∧F,N∨E~N∨F成...本文研究了在von Neumann代数框架下,投影运算后等价的保持情形,特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设E、F、N为von Neumann代数ℳ中的三个有限投影,且E~F。当NE=NF=0时,那么有N+E~N+F,N∧E~N∧F,N∨E~N∨F成立。当NE,NF时,那么有N∧E~N∧F,N∨E~N∨F成立。当EN,FN时,那么有N−E~N−F,N∧E~N∧F,N∨E~N∨F成立。本文进一步推至N为von Neumann代数ℳ中的无限投影,并且考虑了von Neumann代数ℳ中四个投影运算的情况。This paper studies the conditions of equivalents after projection operations under the framework of von Neumann algebra, especially the characterization and properties of projection equivalents in different types of von Neumann algebras. Let E,F,Nbe three finite projections in von Neumann algebra ℳ, and E~F. If NE=NF=0, then it follows that N+E~N+F, N∧E~N∧F, N∨E~N∨Fholds. If NE, NF, then it follows that N∧E~N∧F, N∨E~N∨Fholds. If EN, FN, then it follows that N−E~N−F, N∧E~N∧F, N∨E~N∨Fholds. This paper further extends these results to infinite projections Nin von Neumann algebras ℳand considers the case of four projection operations within von Neumann algebras ℳ.展开更多
设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使...设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使得Φ(A)=Φ(I)Ψ(A),∀A∈ℳ,其中Φ(I)∈N是可逆中心元且Φ(I)2=I′。该结论将因子von Neumann代数上的非线性混合Lie可乘双射的结果推广到无I1或I2型中心直和项的von Neumann代数。Let ℳand Nbe von Neumann algebras with no central summands of type I1or I2, Iand I′be the identities of them. This paper proves that a bijective map Φ:ℳ→Nis mixed Lie multiplicative, that is, Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳif and only if Φ(A)=Φ(I)Ψ(A)for all A∈ℳ, where Ψ:ℳ→Nis a direct sum of a linear *-isomorphism and a conjugate linear *-isomorphism, Φ(I)is a central element in Nwith Φ(I)2=I′. The results about mixed Lie multiplicative maps on factor von Neumann algebras are generalized to von Neumann algebras with no central summands of type I1or I2.展开更多
本文在有限von Neumann代数的情形下应用广义奇异值的方法证明了一类迹函数的若干性质。特别地,我们将Hansen的主要结果推广至有限von Neumann代数的情形。In this paper, via the method of generalized singular values, we prove som...本文在有限von Neumann代数的情形下应用广义奇异值的方法证明了一类迹函数的若干性质。特别地,我们将Hansen的主要结果推广至有限von Neumann代数的情形。In this paper, via the method of generalized singular values, we prove some properties of a class of trace functions defined over finite von Neumann algebras. In particular, we extend the main results of Hansen to the context of finite von Neumann algebras.展开更多
文摘Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I.For all A,B∈M,define by A■B=AB+BA the Jordan product of A and B,A·_(η)B=AB+ηBA^(*)the Jordan η-*-product of A and B,respectively.Let Φ and Φ^(-1)preserve the mixed Jordan triple η-*-products.It is proved that Φ is a linear *-isomorphism if η is not real and Φ is the sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η is real.
文摘本文研究了在von Neumann代数框架下,投影运算后等价的保持情形,特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设E、F、N为von Neumann代数ℳ中的三个有限投影,且E~F。当NE=NF=0时,那么有N+E~N+F,N∧E~N∧F,N∨E~N∨F成立。当NE,NF时,那么有N∧E~N∧F,N∨E~N∨F成立。当EN,FN时,那么有N−E~N−F,N∧E~N∧F,N∨E~N∨F成立。本文进一步推至N为von Neumann代数ℳ中的无限投影,并且考虑了von Neumann代数ℳ中四个投影运算的情况。This paper studies the conditions of equivalents after projection operations under the framework of von Neumann algebra, especially the characterization and properties of projection equivalents in different types of von Neumann algebras. Let E,F,Nbe three finite projections in von Neumann algebra ℳ, and E~F. If NE=NF=0, then it follows that N+E~N+F, N∧E~N∧F, N∨E~N∨Fholds. If NE, NF, then it follows that N∧E~N∧F, N∨E~N∨Fholds. If EN, FN, then it follows that N−E~N−F, N∧E~N∧F, N∨E~N∨Fholds. This paper further extends these results to infinite projections Nin von Neumann algebras ℳand considers the case of four projection operations within von Neumann algebras ℳ.
文摘设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使得Φ(A)=Φ(I)Ψ(A),∀A∈ℳ,其中Φ(I)∈N是可逆中心元且Φ(I)2=I′。该结论将因子von Neumann代数上的非线性混合Lie可乘双射的结果推广到无I1或I2型中心直和项的von Neumann代数。Let ℳand Nbe von Neumann algebras with no central summands of type I1or I2, Iand I′be the identities of them. This paper proves that a bijective map Φ:ℳ→Nis mixed Lie multiplicative, that is, Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳif and only if Φ(A)=Φ(I)Ψ(A)for all A∈ℳ, where Ψ:ℳ→Nis a direct sum of a linear *-isomorphism and a conjugate linear *-isomorphism, Φ(I)is a central element in Nwith Φ(I)2=I′. The results about mixed Lie multiplicative maps on factor von Neumann algebras are generalized to von Neumann algebras with no central summands of type I1or I2.
文摘本文在有限von Neumann代数的情形下应用广义奇异值的方法证明了一类迹函数的若干性质。特别地,我们将Hansen的主要结果推广至有限von Neumann代数的情形。In this paper, via the method of generalized singular values, we prove some properties of a class of trace functions defined over finite von Neumann algebras. In particular, we extend the main results of Hansen to the context of finite von Neumann algebras.
