Generalized duality and product of some noncommutative symmetric spaces (Q2831248)

Let \(\mathcal{M}\) be a semifinite von Neumann algebra, and let \(E\) and \(F\) be symmetric quasi-Banach function spaces on \(I\), where \(I=(0,1)\) or \((0,\infty)\). The author denotes by \(M(E,F)\) the space of multipliers, i.e., the set \(\{x\in L_0(I): xy\in F \text{ for every } y\in E\}\) with the norm \(\|x\|_{M(E,F)}=\sup \{\|xy\|_F: y\in E, \,\|y\|_E\leq1\}.\) This generalizes the Köthe dual \(E^\times\) of \(E\), equal to \(M(E, L^1).\) Further, \(E\odot F=\{fg: f\in E,\, g\in F\}\) is called the product space.NEWLINENEWLINEThe purpose of the paper is to investigate the notions in the noncommutative setting, where for a symmetric quasi-Banach space \(E\) one considers the symmetric quasi-Banach space \(E(\mathcal{M})=\{x\in L_0(\mathcal{M}): \mu(x)\in E\}.\) It is shown that under some natural convexity conditions on \(E\) and \(F\), the space \(M(E,F)\) is a symmetric quasi-Banach space, and with \(F\) fully symmetric one gets \(M(E(\mathcal{M}), F(\mathcal{M}))=M(E,F)(\mathcal{M}).\) As an application, conditions are given under which \(M(E(\mathcal{M}), F(\mathcal{M}))\odot E(\mathcal{M})=F(\mathcal{M})\), and the product space is described with the help of the complex interpolation method.