This is an announcement for the paper “Unbounded Norm Convergence in Banach Lattices” by Y. Deng, M. O’Brien, V. G. Troitsky.
Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is unbounded order convergent to $x\in X$ if $|x_\alpha-x|\hat u$ converges to 0 in order for all $u\in X_+$. This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $(x_\alpha)$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $||x_\alpha-x|\hat u|\rightarrow 0$ for all $u\in X_+$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1605.03538