Narrow Results

Let $E$ and $F$ be two symmetric quasi-Banach spaces and let ${ℳ}$ be a semifinite von Neumann algebra. The purpose of this paper is to study the product space $E({ℳ})\odot F({ℳ})$ and the space of multipliers from $E({ℳ})$ to $F({ℳ})$, i.e. $M(E({ℳ}),F({ℳ}))$. These spaces share many properties with their classical counterparts. Let $0<{\alpha }_0,{\alpha }_1<\infty .$ It is shown that if $F$ is ${\alpha }_1$-convex fully symmetric and $E$ is ${\alpha }_0$-convex, then $M(E({ℳ}),F({ℳ}))=M(E,F)({ℳ})$, where $M(E,F)({ℳ})=\{x\in L_0({ℳ}):\mu (x)\in M(E,F)\}$ and $M(E,F)$ is the space of multipliers from $E$ to $F.$ As an application, we give conditions on when $M(E({ℳ}),F({ℳ}))\odot E({ℳ})=F({ℳ}).$ Moreover, we show that the product space can be described with the help of complex interpolation method.

We introduce the notion of a duality between compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs, which is a generalization of that of the duality between compact/non-compact Riemannian symmetric pairs. Moreover we state a relation between compact symmetric triads and symmetric triads with multiplicities. As its application we state an outline of an alternative proof of Berger’s classification of pseudo-Riemannian symmetric pairs. Moreover, we also give an alternative proof of Leung’s classification of reflective submanifolds in compact Riemannian symmetric spaces and that of real forms in compact Hermitian symmetric spaces.