Natural bilinear forms, natural sesquilinear forms and the associated duality on non-commutative L^p-spaces

In the author's previous paper, he constructed a complex one-parameter family of non-commutative L^p-spaces L^p_(α)(φ), α ∈ ℂ, 1 < p < ∞, for a von Neumann algebra M with respect to a fixed faithful normal semi-finite weight φ on M by using Calderón's complex interpolation method. In this paper, we will construct bounded non-degenerate bilinear forms > , <_p,(α) on L^p_(α)(φ) × L^q_(-α)(φ), α ∈ ℂ, 1 < p < ∞, 1/p + 1/q = 1, and bounded non-degenerate sesquilinear forms (α)_p,(α) on L^p_(α)(φ) × L^q_(ᾱ)(φ), α ∈ ℂ, 1 < p < ∞, 1/p + 1/q = 1, and by using general theory of the complex interpolation method we show the reflexivity of L^p_(α)(φ) and the duality between L ^p_(α)(φ) and L^q_(-α)(φ) via > , <_p,(α) (or the duality between L^p_(α)(φ) and L^q_(ᾱ)(φ) via (α)_p,(α)). Moreover, we discuss bimodule properties of L^p_(α)(φ).