On bounded coordinates in GNS spaces
Volume 583 / 2023
Abstract
We provide a comprehensive study of uniformly left bounded $($resp. left-right bounded$)$ orthonormal bases in GNS spaces of infinite-dimensional von Neumann algebras in the framework of both faithful normal states and f.n.s. weights. There are two issues to consider: one concerning the existence of such bases and the other concerning the bound in operator norm of the left $($resp. left and right$)$ multiplication operators associated to such bases. We provide necessary and sufficient conditions on a closed subspace of a GNS space to guarantee the existence of an orthonormal basis of uniformly left bounded $($resp. left-right bounded$)$ vectors. In the context of states, while a basis of the first kind exists for all GNS spaces, $\mathbf{B}(\ell^2)$ is excluded for a basis of the latter kind. However, in the context of weights, there are no such obstructions. In the context of weights, the GNS space of every infinite-dimensional von Neumann algebra admits a uniformly left and right bounded orthonormal basis such that the aforesaid bound is arbitrarily small.
If $M$ is an infinite-dimensional factor and $\varphi$ is a faithful normal state on $M$, then given $\epsilon \gt 0$, the associated GNS space admits a uniformly left bounded orthonormal basis $\mathcal{O}$ such that $\sup_{\xi\in\mathcal{O}}\|{L_\xi}\|\leq (1+\sqrt{2})+\epsilon$.
If $M$, $\varphi$ and $\epsilon$ are as above, and $M$ is either of type $\mathrm{II}$ or $\mathrm{III}_\lambda$ with $\lambda\in [0,1)$, then the GNS space of $\varphi$ admits a left and right bounded orthonormal basis $\mathcal{O}$ such that \begin{align*} \sup_{\xi\in\mathcal{O}} \max(\|{L_\xi}\|,\|{R_\xi}\|)\leq (1+\sqrt{2})+\epsilon. \end{align*} Similar is the conclusion if $M$ is of type $\mathrm{III}_1$ and $\varphi$ is almost periodic. If $\varphi$ is not tracial and $\mathcal{O}$ is a uniformly left and right bounded orthonormal basis as stated above, there exists $\delta \gt 0$ such that \begin{align*} 1+\delta\leq\sup_{\xi\in\mathcal{O}}\max(\|{L_\xi}\|,\|{R_\xi}\|)\leq(1+\sqrt{2})+\epsilon. \end{align*} Related questions on unitary bases remain open and untouched since the 1967 Baton Rouge conference.
