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Register a new userStrong sums of projections in type ${rm II}$ factors
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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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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.
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.
We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.
Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]sqsubseteq [q]$ iff for every primitive ideal $mathfrak p$ of $A$ either $p/mathfrak ppreceq q/mathfrak ppreceq (1- q)/mathfrak p$ or $p/mathfrak psucceq q/mathfrak p succeq (1-q)/mathfrak p.$ We prove that $p$ is central iff $[p]$ is $sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]mapsto [p]^Game$ that moves $p$ towards the center. The number $n(p) $ of $Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]sqsubseteq [q]Rightarrow n(p)leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliotts classification, the categorical equivalence $Gamma$ between MV-algebras and unital $ell$-groups, and L os ultraproduct theorem for first-order logic.
It is shown that for any approximately central (AC) projection $e$ in the Flip orbifold $A_theta^Phi$ (of the irrational rotation C*-algebra $A_theta$), and any modular automorphism $alpha$ (arising from SL$(2,mathbb Z)$), the AC projection $alpha(e)$ is centrally Murray-von Neumann equivalent to one of the projections $e, sigma(e), kappa(e), kappa^2(e),$ $sigmakappa(e), sigmakappa^2(e)$ in the $S_3$-orbit of $e,$ where $sigma, kappa$ are the Fourier and Cubic transforms of $A_theta$. (The equivalence being implemented by an approximately central partial isometry in $A_theta^Phi$.) For smooth automorphisms $alpha,beta$ of the Flip orbifold $A_theta^Phi$, it is also shown that if $alpha_*=beta_*$ on $K_0(A_theta^Phi),$ then $alpha(e)$ and $beta(e)$ are centrally equivalent for each AC projection $e$.
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