$L^{1}$ representation of Riesz spaces

Bahri Turan¹

¹ Department of Mathematics Faculty of Arts and Science Gazi University Teknikokullar 06500 Ankara, Turkey

Studia Mathematica, Tome 176 (2006) no. 1, pp. 61-68

Résumé

Let $E$ be a Riesz space. By defining the spaces $L_{E}^{1}$ and $L_{E}^{\infty }$ of $E$, we prove that the center $Z(L_{E}^{1})$ of $L_{E}^{1}$ is $L_{E}^{\infty }$ and show that the injectivity of the Arens homomorphism $m:Z(E)^{\prime \prime }\rightarrow Z(E^{\sim })$ is equivalent to the equality $L_{E}^{1}=Z(E)^{\prime }$. Finally, we also give some representation of an order continuous Banach lattice $E$ with a weak unit and of the order dual $E^{\sim }$ of $E$ in $L_{E}^{1}$ which are different from the representations appearing in the literature.

Détail

DOI : 10.4064/sm176-1-4

Keywords: riesz space defining spaces infty prove center infty injectivity arens homomorphism prime prime rightarrow sim equivalent equality prime finally representation order continuous banach lattice weak unit order dual sim which different representations appearing literature