I am reading some paper about L-embedded space. For the definition of L-embedded space, see http://www.sciencedirect.com/science/article/pii/S0022247X02001075.
Let $Y$ be a Banach space and $P$ a projection on $Y$. $P$ is called an L-projection if $x= Px + (I_Y −P)x$ for all $x ∈ Y$. A closed subspace $X ⊂ Y$ is called an L-summand on $Y$ if $X$ is the range of an L-projection on $Y$. When $Y=X^{**}$(the bidule of $X$), it is said that $X$ is an L-embedded Banach space and there exists a closed subspace $X_s\subset X^{**}$ such that $X^{**} = X\oplus X_s$.
My question is: it seems every $L_p$ space is L-embedded, but no paper mention that $L_p$, $1<p<\infty$, is L-embedded, only found $L_1$ is L-embedded. I think if a space is reflexive, then $X_s=\emptyset$ and the L-embedded projection is just identity.
That's right. Reflexive spaces are both $M$-embedded and $L$-embedded.