保零积映射的刻画及其应用
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摘要
算子理论与算子代数近几十年来的发展表明,对算子代数上保持某些同构不变量的映射的刻画和分类问题研究有助于加深人们对算子代数结构的了解([1]).这些不变量包括算子间的关系,算子的集合和函数等.由于零积关系是代数中广泛存在的基本重要的关系,对于保持算子零积关系的映射的刻画问题尤被人们关注,成为被广泛深入研究的课题之一.设(?)是算子的某种乘积.算子A,B间具有零积关系,即满足乘积A(?)B等于零.算子代数上的映射Φ满足A(?) B=0(?)Φ(A)(?)Φ(B)=0,则称映射Φ(双边)保零积.对于算子代数上保零积映射刻画问题的研究成果不仅揭示了算子代数的某些固有的结构性质,而且也被广泛的应用于其他研究领域.本文研究某些算子代数或算子集合上双边保零积非线性映射的刻画问题,及其在其他相关领域,特别是量子物理学中的应用.下面是本文的主要结果.
1.系统地讨论了几类算子集合上双边保零积的非线性映射的结构特征和刻画问题.设X,H分别是实或复数域F上的Banach空间或Hilbert空间,其维数均大于等于3.ω,ν(?)B(X)或者B(H),映射Φ:ω→ν满足A(?) B=0(?)Φ(A)(?)Φ(B)=0对所有A,B∈ω成立.我们分别在下列几种情形给出了上述映射在秩一算子上的形式：(1)ω,ν(?)B(x)为包含秩一幂等元的算子集合,垂是满射且A(?)B=AB;(2)ω,ν为H上包含秩一投影的自伴算子集合,Φ是满射且A(?)B=AB;(3)w,ν(?)B(H)为包含秩一投影的自伴算子集合,Φ是双射且A(?)B=AB+和A+B或者A(?)B=AB+A(设J∈B(H)是可逆自伴算子,A+=J-1A*J)；(4)ω=ν=Cp(H),Φ是实线性满射且取A(?)B=0为A*B=AB*=0; (5) W, V(?)B(H)为包含秩一投影的正算子集合,中是双射且取AoB=ATB(这里T是任意给定的正可逆算子).
2.利用上述对算子集合上双边保零积映射的刻画结果,我们给出了相应算子集合上保相应乘积数值半径或交叉范数映射的完全刻画.
3.利用算子集合上双边保正交性映射的刻画结果,我们得到了Schatten-p类算子空间上的保距映射和完全保距映射的刻画,以及套代数中Schatten-p类算子上保距映射的完全刻画.
4.利用包含秩一投影的正算子集合上双边保算子广义正交性映射的刻画结果,给出了量子力学基本定理之一Wigner定理的一类推广.
5.利用包含秩一投影的正算子集合上双边保算子正交性映射的刻画结果,给出了Hilbert空间效应代数上一般的序列同构的刻画.设H是复Hilbert空间,dimH≥2,令运算*是Hilbert空间效应代数ε(H)上的任一序列乘积,双射Φ:ε(H)→ε(H)是序列同构,即满足Φ(A*B)=Φ(A)*Φ(B)对所有A,B∈ε(H)都成立,则存在酉或反酉算子U使得Φ(A)=UAU*对所有A∈ε(H)都成立.
The study on the problem of characerizing maps on operator algebras that preserve some invariants of isomorphisms is helpful for making the structure of operator algebras to be understood better ([1]). Such invariants include some relations between operators, some sets of operators and some functions on operators, etc.. Especially, since the zero product is a very basic relation, many authors have been payed attention to the maps preserving zero products of operators. For a product operation o defined in operator algebras, a mapΦpreserves zero products (in both directions) if A(?)B=0(?)((?) )Φ(A)(?)Φ(B)=0. Surprisedly, the results on characterizing zero product preserving maps not only the research help us understander the structure of operator algebras better, but also are applied to other research fields extensively. In the present thesis, we devote to characterizing maps on several operator sets preserving zero products in both directions, and applying these results to the research of other topics, such as the general preserver problems and some problems in Quantum Mechanics. The following are the main results obtained in this thesis:
1. We characterize the following kinds of maps preserving zero products in both directions. Let X and H be a Banach space and a Hilbert space on the real or complex field F respectively with dim X≥3 and dim H≥3. Let W, V(?) B(X) or B(H) be two subsets andΦ:W→V be a map satisfies A(?)B=0(?)Φ(A)(?)Φ(B)=0 for all A, B∈W. We give a characterization of the above maps for the following cases:(1) W, V (?)B(X) are subsets containing all idempotences of rank one,Φis asurjection and A(?)B=AB; (2) W, V (?)BS(H)(BS(H) is the space of self adjoint operators on H) are subsets containing all projections of rank one,Φis a surjection and A(?)B=AB; (3) W,V(?)B(H) are subsets containing all operators of rank one,Φis a bijection and A(?)B=AB(?) and A(?)B or A(?)B=AB(?)A, where J∈B(H) is a self-adjoint invertible operator, A(?)= J-1A*J; (4) W=V=CP(H),Φis a real linear surjective map and AοB=0 is replaced by A*B=AB*=0; (5) W,V (?) B(H) are subsets of positive operators containing all projections of rank one,Φis bijective and A(?)B= ATB (T is an arbitrarily given positive invertible operator).
2. Applying the results in 1, we obtain characterizaions of maps on corresponding operator sets preserving numerical radius or cross norms of corresponding products of operators.
3. Applying the result 1(4) of characerizing maps preserving orthogonality in both directions on the operator sets, we give a characterizaion of distance preserving and com-pletely distance preserving maps on the Schatten-p class, and distance preserving maps on the Schatten-p class in nest algebras.
4. Applying the result of classificating maps preserving generalized orthogonality in both directions on the set of positive operators contains rank one projections, we give a generalization of Wigner theorem in theory of Quantum Mechanics.
5. Also applying the result of classificating maps preserving orthogonality in both directions on the set of positive operators containing rank one projections, we give a characterization of general sequential isomorphisms on Hilbert space effect algebras. Let the operation (?) be an arbitrary sequential product on the Hilbert space effect algebrasε(H) on a Hilbcrt space H with dimH≥2. If the mapΦ:ε(H)→ε(H) is bijective and satisfies thatΦ(A(?)B)=Φ(A)(?)(B) for A, B∈ε(H), then there is an either unitary or anti-unitary operator U such thatΦ(A)=UAU* for every A∈ε(H).
1.系统地讨论了几类算子集合上双边保零积的非线性映射的结构特征和刻画问题.设X,H分别是实或复数域F上的Banach空间或Hilbert空间,其维数均大于等于3.ω,ν(?)B(X)或者B(H),映射Φ:ω→ν满足A(?) B=0(?)Φ(A)(?)Φ(B)=0对所有A,B∈ω成立.我们分别在下列几种情形给出了上述映射在秩一算子上的形式：(1)ω,ν(?)B(x)为包含秩一幂等元的算子集合,垂是满射且A(?)B=AB;(2)ω,ν为H上包含秩一投影的自伴算子集合,Φ是满射且A(?)B=AB;(3)w,ν(?)B(H)为包含秩一投影的自伴算子集合,Φ是双射且A(?)B=AB+和A+B或者A(?)B=AB+A(设J∈B(H)是可逆自伴算子,A+=J-1A*J)；(4)ω=ν=Cp(H),Φ是实线性满射且取A(?)B=0为A*B=AB*=0; (5) W, V(?)B(H)为包含秩一投影的正算子集合,中是双射且取AoB=ATB(这里T是任意给定的正可逆算子).
2.利用上述对算子集合上双边保零积映射的刻画结果,我们给出了相应算子集合上保相应乘积数值半径或交叉范数映射的完全刻画.
3.利用算子集合上双边保正交性映射的刻画结果,我们得到了Schatten-p类算子空间上的保距映射和完全保距映射的刻画,以及套代数中Schatten-p类算子上保距映射的完全刻画.
4.利用包含秩一投影的正算子集合上双边保算子广义正交性映射的刻画结果,给出了量子力学基本定理之一Wigner定理的一类推广.
5.利用包含秩一投影的正算子集合上双边保算子正交性映射的刻画结果,给出了Hilbert空间效应代数上一般的序列同构的刻画.设H是复Hilbert空间,dimH≥2,令运算*是Hilbert空间效应代数ε(H)上的任一序列乘积,双射Φ:ε(H)→ε(H)是序列同构,即满足Φ(A*B)=Φ(A)*Φ(B)对所有A,B∈ε(H)都成立,则存在酉或反酉算子U使得Φ(A)=UAU*对所有A∈ε(H)都成立.
The study on the problem of characerizing maps on operator algebras that preserve some invariants of isomorphisms is helpful for making the structure of operator algebras to be understood better ([1]). Such invariants include some relations between operators, some sets of operators and some functions on operators, etc.. Especially, since the zero product is a very basic relation, many authors have been payed attention to the maps preserving zero products of operators. For a product operation o defined in operator algebras, a mapΦpreserves zero products (in both directions) if A(?)B=0(?)((?) )Φ(A)(?)Φ(B)=0. Surprisedly, the results on characterizing zero product preserving maps not only the research help us understander the structure of operator algebras better, but also are applied to other research fields extensively. In the present thesis, we devote to characterizing maps on several operator sets preserving zero products in both directions, and applying these results to the research of other topics, such as the general preserver problems and some problems in Quantum Mechanics. The following are the main results obtained in this thesis:
1. We characterize the following kinds of maps preserving zero products in both directions. Let X and H be a Banach space and a Hilbert space on the real or complex field F respectively with dim X≥3 and dim H≥3. Let W, V(?) B(X) or B(H) be two subsets andΦ:W→V be a map satisfies A(?)B=0(?)Φ(A)(?)Φ(B)=0 for all A, B∈W. We give a characterization of the above maps for the following cases:(1) W, V (?)B(X) are subsets containing all idempotences of rank one,Φis asurjection and A(?)B=AB; (2) W, V (?)BS(H)(BS(H) is the space of self adjoint operators on H) are subsets containing all projections of rank one,Φis a surjection and A(?)B=AB; (3) W,V(?)B(H) are subsets containing all operators of rank one,Φis a bijection and A(?)B=AB(?) and A(?)B or A(?)B=AB(?)A, where J∈B(H) is a self-adjoint invertible operator, A(?)= J-1A*J; (4) W=V=CP(H),Φis a real linear surjective map and AοB=0 is replaced by A*B=AB*=0; (5) W,V (?) B(H) are subsets of positive operators containing all projections of rank one,Φis bijective and A(?)B= ATB (T is an arbitrarily given positive invertible operator).
2. Applying the results in 1, we obtain characterizaions of maps on corresponding operator sets preserving numerical radius or cross norms of corresponding products of operators.
3. Applying the result 1(4) of characerizing maps preserving orthogonality in both directions on the operator sets, we give a characterizaion of distance preserving and com-pletely distance preserving maps on the Schatten-p class, and distance preserving maps on the Schatten-p class in nest algebras.
4. Applying the result of classificating maps preserving generalized orthogonality in both directions on the set of positive operators contains rank one projections, we give a generalization of Wigner theorem in theory of Quantum Mechanics.
5. Also applying the result of classificating maps preserving orthogonality in both directions on the set of positive operators containing rank one projections, we give a characterization of general sequential isomorphisms on Hilbert space effect algebras. Let the operation (?) be an arbitrary sequential product on the Hilbert space effect algebrasε(H) on a Hilbcrt space H with dimH≥2. If the mapΦ:ε(H)→ε(H) is bijective and satisfies thatΦ(A(?)B)=Φ(A)(?)(B) for A, B∈ε(H), then there is an either unitary or anti-unitary operator U such thatΦ(A)=UAU* for every A∈ε(H).
引文
[1]侯晋川,崔建莲,算子代数上的线性映射引论.科学出版社,北京,2002.
[2]Cui J, Hou J. Maps preserving functional values of operator products invariant, Linear Algebra Appl., 428 (2008),1649-1663.
[3]Chan J, Li C, Sze N. Mappings on matrices:Invariance of functional values of matrix products, J. Austral. Math. Soc. (Serie A),81 (2006),165-184.
[4]Dobovisek M, Kuzma B, Lesnjak G, Li C., Petek T. Mapping that preserves pairs of operators with zero triple jordan product, Linear Algebra Appl.,426(2007),255-279
[5]Molnar L. A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules, J. Math. Phys.,40 (1999),5544-5554.
[6]Molnar L. A Wigner-type theorem on symmetry transformations in type Ⅱ factor, International Journal of Theoretical Physics,39 (2000),1463-1466.
[7]Molnar L. Maps on the n-dimensional subspaces of a Hilbert space preserving principal angles, Proc. Amer. Math. Soc.,136 (2008),3205-3209.
[8]Molnar L, Timmermann W. Isometries of quantum states, J. Phys. A:Math. Gen.,36 (2003), 267-273.
[9]Molnar L. Orthogonality preserving transformations on indefinite inner product space:Gen-eralization of Uhlhorn's version of Wigner's theorem, J. Funct. Anal.,194(2002),248-262.
[10]Molnar L. Preservers on Hilbert space effect algebras, Linear Algebra Appl.,370(2003),287-300.
[11]Molnar L. On some automorphisms of the set of efects on Hilbert space, Lett. Math. Phys. 51 (2000),37-45.
[12]Semrl P. Applying projective geometry to trasformations on rank one idempotents, J. Func. Anal.,210(2004),248-257.
[13]Semrl P. Non-linear communitavity preserving maps, preprint.
[14]Semrl P. Maps on idempotents, Studia Math.,169 (2005),21-44.
[15]Cui J, Hou J. Linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest, Int. Equ. Oper. Theo.,46 (2003),253-266.
[16]Chan J. Numerical radius preserving operators on B(H), Proc. Amer. Math. Soc.,123 (1995), 1437-1439.
[17]Chan J. Numerical radius preserving operators on C*-algebras, Arch. Math. (Basel),70 (1998), 486-488.
[18]Li C, Tsing N. linear preservers on numerical ranges, numerical radii and unitary similarity invariant norms, Linear and Multilinear Algebra,33 (1992),63-73.
[19]Bai Z, Hou J. Numerical radius distance preserving maps on B(H), Proc. Amer. Math. Soc. 132 (2004),1453-1461.
[20]Bai Z, Hou J., Xu Z. Maps preserving numerical radius on C*-algebras, Studia Math.,162 (2004),97-104.
[21]Cui J, Hou J. Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras, J. Funct. Anal.,206 (2004),414-448.
[22]Hou J, Di Q. Maps preserving numerical range of operator products, Proc. Amer. Math. Soc., 134 (2006),1435-1446.
[23]Li C, Sze N. Product of Operators and Numerical Range Preserving Maps, Studia Math.,174 (2006),169-182.
[24]Guo H, Li C. C*-Isomorphisms, Jordan Isomorphisms, and Numerical Range Preserving Maps, Proc. Amer. Math. Soc.,135 (2007),2907-2914.
[25]Arazy J. The isometries of Cp. Israel J. Math.22 (1975),247-256.
[26]Erdos J. A simple proof of Arazy's theorem. Proc.Edinburgh Math. Soc.37(1994),239-242.
[27]Kadison R. Isometries of operator algebras. Ann. of Math. (2) 54 (1951),325-338.
[28]Noussis M, Katavolos A. Isometries of Cp spaces in nest algebras. J. London. Math. Soc.(2) 51 (1995),175-188.
[29]Effros E, Ruan Z. Operator spaces, London Math. Soc. Monographs, New Series 23, Oxford University Press. New York 2000.
[30]Ruan Z. A classification for 2-isometries of noncommunative Lp-space. Israel J. Math. 150(2005),285-314.
[31]Bai Z, Hou J, Du S. Rank one preserving addtivie maps in nest algebras. Linear and Multilinear Algebra,30(2009), inpress.
[32]Bai Z, Hou J. Distance preserving maps in nest algebras. preprint.
[33]Uhlhorn U. Representation of symmetry transformations in quantum mechanics, Ark. Fysik, 23 (1963),307-340.
[34]Broek P. Group representations in indefinite metric spaces. J. Math. Phys.,25 (1984),1205-1210.
[35]Broek P. Symmetry transformations in indefinite metric spaces:A generalization of Wigner's theorem, Physics A,127(1984),599-612.
[36]Schreckenberg S. Symmetry and history quantum theory:An analog of Wigner's theorem, J. Math. Phys.,37(1996),6086-6105.
[37]Bengtsson I, Zyczkowski K. Geometry of Quantum States:An Introduction to Quantum Entanglement, Cambridge University Press, Cambridge,2006.
[38]Petz D. Monotone metrics on matrix spaces, Linear Algebra Appl.,244(1996),81-96.
[39]Petz D. Geometry of canonical correlation on the state space of a quantum system, J. Math. Phys.,35(1994),780-795.
[40]Ludwig G. Foundations of Quantum Machanics, Vol I, Springer Berlin,1983.
[41]Gudder S. Tensor products of sequential effect algebras. Mathematica Slovaca, preprint.
[42]Gudder S, Greechie R. Sequentially products on effect algebras, Reports on Mathematical Physics 49, (2002),87-111.
[43]Gudder S, Greechie R. Uniqueness and order in sequential effect algebras. International Jour-nal of Theoretical Physics, preprint.
[44]Gudder S,Nagy G. Sequential quantum measurements, J. Math. Phys.42 (2001),5212-5222.
[45]Gudder S, Open Problems for Sequential Effect Algebras, International Journal of Theoretical Physics, inpress.
[46]Liu W,Wu J. A uniqueness problem of the sequence product on Hilbert space effect algebra, J. Phys. A:Math. Theor.42(2009),185206.
[47]Morozova E, Chentsov N. Markov invariant geometry on state manifolds (in Russian), Itogi Nauki i Tekhniki,36 (1990) 69-102.
[48]Mazur S, Ulam S. Sur les transformations isometriques d'espaces vectoriels normes. C. R. Acad. Sci. Paris 194 (1932),946-948.
[49]Shen J, Wu J. Sequential product on standard effect algebra,J. Phys. A:Math. Theor.42 (2009) 345203.
[50]Cui J. Hou J, Park C. Indefinite orthogonality preserving additive maps, Arch. Math.83 (2004) 548-557.
[51]Faure C. An elementary proof of the fundamental theorem of projective geometry. Geom. Dedicata 90 (2002) 145-151.
[52]Kubo F, Ando T. Means of positive linear operators, Ann. of Math.,246 (1980),205-224.
[53]McCarthy C. Cp. Israel J. Math.5 (1967),249-271.
[54]Moore L, Trent T. Isometries on nest algebras, J.Func.Anal.,86(1989),180-209.
[55]Ruan Z. Subspaces of C*-algebras. J. Funct. Anal.76 (1988),217-230.
[56]Russo B. Isometries of the trace class. Proc. Amer. Math. Soc.23 (1969),213.
[57]Semrl P. Maps on idempotent operators, Banach Center Publications,75 (2007),289-301.
[2]Cui J, Hou J. Maps preserving functional values of operator products invariant, Linear Algebra Appl., 428 (2008),1649-1663.
[3]Chan J, Li C, Sze N. Mappings on matrices:Invariance of functional values of matrix products, J. Austral. Math. Soc. (Serie A),81 (2006),165-184.
[4]Dobovisek M, Kuzma B, Lesnjak G, Li C., Petek T. Mapping that preserves pairs of operators with zero triple jordan product, Linear Algebra Appl.,426(2007),255-279
[5]Molnar L. A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules, J. Math. Phys.,40 (1999),5544-5554.
[6]Molnar L. A Wigner-type theorem on symmetry transformations in type Ⅱ factor, International Journal of Theoretical Physics,39 (2000),1463-1466.
[7]Molnar L. Maps on the n-dimensional subspaces of a Hilbert space preserving principal angles, Proc. Amer. Math. Soc.,136 (2008),3205-3209.
[8]Molnar L, Timmermann W. Isometries of quantum states, J. Phys. A:Math. Gen.,36 (2003), 267-273.
[9]Molnar L. Orthogonality preserving transformations on indefinite inner product space:Gen-eralization of Uhlhorn's version of Wigner's theorem, J. Funct. Anal.,194(2002),248-262.
[10]Molnar L. Preservers on Hilbert space effect algebras, Linear Algebra Appl.,370(2003),287-300.
[11]Molnar L. On some automorphisms of the set of efects on Hilbert space, Lett. Math. Phys. 51 (2000),37-45.
[12]Semrl P. Applying projective geometry to trasformations on rank one idempotents, J. Func. Anal.,210(2004),248-257.
[13]Semrl P. Non-linear communitavity preserving maps, preprint.
[14]Semrl P. Maps on idempotents, Studia Math.,169 (2005),21-44.
[15]Cui J, Hou J. Linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest, Int. Equ. Oper. Theo.,46 (2003),253-266.
[16]Chan J. Numerical radius preserving operators on B(H), Proc. Amer. Math. Soc.,123 (1995), 1437-1439.
[17]Chan J. Numerical radius preserving operators on C*-algebras, Arch. Math. (Basel),70 (1998), 486-488.
[18]Li C, Tsing N. linear preservers on numerical ranges, numerical radii and unitary similarity invariant norms, Linear and Multilinear Algebra,33 (1992),63-73.
[19]Bai Z, Hou J. Numerical radius distance preserving maps on B(H), Proc. Amer. Math. Soc. 132 (2004),1453-1461.
[20]Bai Z, Hou J., Xu Z. Maps preserving numerical radius on C*-algebras, Studia Math.,162 (2004),97-104.
[21]Cui J, Hou J. Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras, J. Funct. Anal.,206 (2004),414-448.
[22]Hou J, Di Q. Maps preserving numerical range of operator products, Proc. Amer. Math. Soc., 134 (2006),1435-1446.
[23]Li C, Sze N. Product of Operators and Numerical Range Preserving Maps, Studia Math.,174 (2006),169-182.
[24]Guo H, Li C. C*-Isomorphisms, Jordan Isomorphisms, and Numerical Range Preserving Maps, Proc. Amer. Math. Soc.,135 (2007),2907-2914.
[25]Arazy J. The isometries of Cp. Israel J. Math.22 (1975),247-256.
[26]Erdos J. A simple proof of Arazy's theorem. Proc.Edinburgh Math. Soc.37(1994),239-242.
[27]Kadison R. Isometries of operator algebras. Ann. of Math. (2) 54 (1951),325-338.
[28]Noussis M, Katavolos A. Isometries of Cp spaces in nest algebras. J. London. Math. Soc.(2) 51 (1995),175-188.
[29]Effros E, Ruan Z. Operator spaces, London Math. Soc. Monographs, New Series 23, Oxford University Press. New York 2000.
[30]Ruan Z. A classification for 2-isometries of noncommunative Lp-space. Israel J. Math. 150(2005),285-314.
[31]Bai Z, Hou J, Du S. Rank one preserving addtivie maps in nest algebras. Linear and Multilinear Algebra,30(2009), inpress.
[32]Bai Z, Hou J. Distance preserving maps in nest algebras. preprint.
[33]Uhlhorn U. Representation of symmetry transformations in quantum mechanics, Ark. Fysik, 23 (1963),307-340.
[34]Broek P. Group representations in indefinite metric spaces. J. Math. Phys.,25 (1984),1205-1210.
[35]Broek P. Symmetry transformations in indefinite metric spaces:A generalization of Wigner's theorem, Physics A,127(1984),599-612.
[36]Schreckenberg S. Symmetry and history quantum theory:An analog of Wigner's theorem, J. Math. Phys.,37(1996),6086-6105.
[37]Bengtsson I, Zyczkowski K. Geometry of Quantum States:An Introduction to Quantum Entanglement, Cambridge University Press, Cambridge,2006.
[38]Petz D. Monotone metrics on matrix spaces, Linear Algebra Appl.,244(1996),81-96.
[39]Petz D. Geometry of canonical correlation on the state space of a quantum system, J. Math. Phys.,35(1994),780-795.
[40]Ludwig G. Foundations of Quantum Machanics, Vol I, Springer Berlin,1983.
[41]Gudder S. Tensor products of sequential effect algebras. Mathematica Slovaca, preprint.
[42]Gudder S, Greechie R. Sequentially products on effect algebras, Reports on Mathematical Physics 49, (2002),87-111.
[43]Gudder S, Greechie R. Uniqueness and order in sequential effect algebras. International Jour-nal of Theoretical Physics, preprint.
[44]Gudder S,Nagy G. Sequential quantum measurements, J. Math. Phys.42 (2001),5212-5222.
[45]Gudder S, Open Problems for Sequential Effect Algebras, International Journal of Theoretical Physics, inpress.
[46]Liu W,Wu J. A uniqueness problem of the sequence product on Hilbert space effect algebra, J. Phys. A:Math. Theor.42(2009),185206.
[47]Morozova E, Chentsov N. Markov invariant geometry on state manifolds (in Russian), Itogi Nauki i Tekhniki,36 (1990) 69-102.
[48]Mazur S, Ulam S. Sur les transformations isometriques d'espaces vectoriels normes. C. R. Acad. Sci. Paris 194 (1932),946-948.
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