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Generalized duality and product of some noncommutative symmetric spaces

Yazhou Han

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, P. R. China

https://doi.org/10.1142/S0129167X16500828Cited by:3 (Source: Crossref)

Abstract

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 $E (ℳ) ⊙ F (ℳ)$ 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 $0 < α_{0}, α_{1} < \infty .$ It is shown that if $F$ is $α_{1}$ -convex fully symmetric and $E$ is $α_{0}$ -convex, then $M (E (ℳ), F (ℳ)) = M (E, F) (ℳ)$ , where $M (E, F) (ℳ) = {x \in L_{0} (ℳ) : μ (x) \in M (E, F)}$ and $M (E, F)$ is the space of multipliers from $E$ to $F .$ As an application, we give conditions on when $M (E (ℳ), F (ℳ)) ⊙ E (ℳ) = F (ℳ) .$ Moreover, we show that the product space can be described with the help of complex interpolation method.

Keywords:

AMSC: 46L52, 46L51, 47L25

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