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Register a new userWigner-Type Theorem on transition probability preserving maps in semifinite factors
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The Wigners theorem, which is one of the cornerstones of the mathematical formulation of quantum mechanics, asserts that every symmetry of quantum system is unitary or anti-unitary. This classical result was first given by Wigner in 1931. Thereafter it has been proved and generalized in various ways by many authors. Recently, G. P. Geh{e}r extended Wigners and Moln{a}rs theorems and characterized the transformations on the Grassmann space of all rank-$n$ projections which preserve the transition probability. The aim of this paper is to provide a new approach to describe the general form of the transition probability preserving (not necessarily bijective) maps between Grassmann spaces. As a byproduct, we are able to generalize the results of Moln{a}r and G. P. Geh{e}r.
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Let $mathcal{A}$ and $mathcal{B}$ be two factor von Neumann algebras and $eta$ be a non-zero complex number. A nonlinear bijective map $phi:mathcal Arightarrowmathcal B$ has been demonstrated to satisfy $$phi([A,B]_{*}^{eta}diamond_{eta} C)=[phi(A),phi(B)]_{*}^{eta}diamond_{eta}phi(C)$$ for all $A,B,Cinmathcal A.$ If $eta=1,$ then $phi$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $eta eq 1$ and satisfies $phi(I)=1,$ then $phi$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.
Let $mathcal{M}$ be a type ${rm II_1}$ factor and let $tau$ be the faithful normal tracial state on $mathcal{M}$. In this paper, we prove that given an $X in mathcal{M}$, $X=X^*$, then there is a decomposition of the identity into $N in mathbb{N}$ mutually orthogonal nonzero projections $E_jinmathcal{M}$, $I=sum_{j=1}^NE_j$, such that $E_jXE_j=tau(X) E_j$ for all $j=1,cdots,N$. Equivalently, there is a unitary operator $U in mathcal{M}$ with $U^N=I$ and $frac{1}{N}sum_{j=0}^{N-1}{U^*}^jXU^j=tau(X)I.$ As the first application, we prove that a positive operator $Ain mathcal{M}$ can be written as a finite sum of projections in $mathcal{M}$ if and only if $tau(A)geq tau(R_A)$, where $R_A$ is the range projection of $A$. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if $Xin mathcal{M}$, $X=X^*$ and $tau(X)=0$, then there exists a nilpotent element $Z in mathcal{M}$ such that $X$ is the real part of $Z$. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that let $X_1,cdots,X_nin mathcal{M}$. Then there exist unitary operators $U_1,cdots,U_kinmathcal{M}$ such that $frac{1}{k}sum_{i=1}^kU_i^{-1}X_jU_i=tau(X_j)I,quad forall 1leq jleq n$. This result is a stronger version of Dixmiers averaging theorem for type ${rm II}_1$ factors.
We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.
We investigate generalizations of the Cramer theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.
Let $M$ be a type ${rm II}$ factor and let $tau$ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type ${rm II}_infty$ case and normalized by $tau(I)=1$ in the type ${rm II}_1$ case. Given $Ain M^+$, we denote by $A_+=(A-I)chi_A(1,|A|]$ the excess part of $A$ and by $A_-=(I-A)chi_A(0,1)$ the defect part of $A$. V. Kaftal, P. Ng and S. Zhang provided necessary and sufficient conditions for a positive operator to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal) in type ${rm I}$ and type ${rm III}$ factors. For type ${rm II}$ factors, V. Kaftal, P. Ng and S. Zhang proved that $tau(A_+)geq tau(A_-)$ is a necessary condition for an operator $Ain M^+$ which can be written as the sum of a finite or infinite collection of projections and also sufficient if the operator is diagonalizable. In this paper, we prove that if $Ain M^+$ and $tau(A_+)geq tau(A_-)$, then $A$ can be written as the sum of a finite or infinite collection of projections. This result answers affirmatively a question raised by V. Kaftal, P. Ng and S. Zhang.
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