TITLE: Complexifying a real Banach space and its dual
QUESTION [11 upvotes]: A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) $(a+ib)(x,y)=(ax-by, bx+ay)$ and a norm by (3) $\|(x,y)\|^\mathbb{C}=\sup_{\theta\in [0,2 \pi]}\|\cos(\theta)x+\sin(\theta)y\|$. (That $\|\cdot\|^\mathbb{C}$ is a norm requires a little proving.)
This definition gives us what we want when going from real $C(K)$ or $l_p$, etc., to their complex versions.

I believe I can show that the (complex) Banach dual of $E_\mathbb{C}$ is the complexification of the real dual $E^*$, i.e., $(E_\mathbb{C})^* = (E^*)_\mathbb{c}$.  But my proof is somewhat messy.  This must surely be in the literature somewhere! Can anyone suggest a reference?
How about the converse?  What if we know that $E_\mathbb{C} = V^*$ for some complex Banach space $V$.  Is $E$ the (real) Banach dual of some Banach space?  (Again, a reference would be appreciated).

[Edit Feb. 13, 2013] Based on Bill Johnson's comment below, perhaps I should motivate the question a bit, and revise question 2.  For a compact space $K$, we know that the Banach space $C(K)$ over the real scalars is isometrically the dual of a real Banach space if and only if $C(K)$ over the complex scalars is isometrically the dual of some complex Banach space. A standard proof is particular to the situation, going through hyperstonian spaces $K$ and normal measures. I was wondering if this is just a special case of a more general fact. So here is a revision of 2. 
Q3. Suppose $E$ is a real Banach lattice and the complexification $E_\mathbb{C}$ is a Banach dual space.  Must $E$ then be the dual of some real Banach space?

REPLY [3 votes]: This is a response to the request for references, but not an answer to Q3. I came across a related paper: "Complexifications of real Banach spaces, polynomials and multilinear maps", by Munoz, Sarantopoulos, and Tonge, Studia Math. 134 (1999), 1-33.
They point out that there are many different ways to put a reasonable norm on the algebraic complexification $E\times E$.