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Let \mathcal M be a semifinite factor with a fixed faithful normal semifinite tracial weight \tau such that \tau (I)=\infty. Denote by \mathscr P(\mathcal M,\tau) the set of all projections in \mathcal M and \mathscr P^{\infty }(\mathcal M,\tau)=\{P\in \mathscr P(\mathcal M,\tau): \tau (P)=\tau (I-P)=\infty \}. In this paper, as a generalization of Uhlhorn's theorem, we establish the general form of orthogonality preserving maps on the Grassmann space \mathscr P^{\infty }(\mathcal M,\tau). We prove that every such map on \mathscr P^{\infty }(\mathcal M,\tau) can be extended to a Jordan *-isomorphism \rho of \mathcal M onto \mathcal M. [ABSTRACT FROM AUTHOR]

In this note, for any two orthogonal projections P,Q on a Hilbert space, the characterization of spectrum of anticommutator PQ+QP has been obtained. As a corollary, an alternative proof of the norm formula has been obtained (see Walters S. Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016). [ABSTRACT FROM AUTHOR]