JEDNOSTKA NAUKOWA KATEGORII A+

Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

On bounded coordinates in GNS spaces

Tom 583 / 2023

Debabrata De, Kunal Mukherjee Dissertationes Mathematicae 583 (2023), 1-106 MSC: Primary 46L10; Secondary 46L36. DOI: 10.4064/dm853-11-2022 Opublikowany online: 15 March 2022

Streszczenie

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.

Autorzy

  • Debabrata DeSchool of Mathematical Sciences
    National Institute of Science Education and Research, Bhubaneswar
    An OCC of Homi Bhabha National Institute
    Jatni 752050, India
    e-mail
  • Kunal MukherjeeDepartment of Mathematics
    IIT Madras
    Chennai 600036
    Tamil Nadu, India
    e-mail

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek