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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.

  1. 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?

  2. 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?

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Isn't $E_{\mathbb{C}}$ just $E \otimes_{\mathbb{R}} \mathbb{C}$ with the injective norm? Do you know Grothendieck's work on topological vector spaces? –  Martin Brandenburg Feb 11 '13 at 2:35
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I believe the standard way is due to Dieudonné: dx.doi.org/10.1090/S0002-9939-1952-0047252-8 where he also proves that the James space is not the underlying real space of a complex Banach space thus disproving a conjecture of Banach. I think Ivan Singer's Bases in Banach spaces, I contains a discussion of the complexification in quite some detail on the first few pages. –  Martin Feb 11 '13 at 2:58
    
Tensor product seems related, but I don't see right now. Thanks to Martin for reference to Dieudonne and to Singer's book. But I still don't see an answer to my question 2 above. –  Fred Dashiell Feb 11 '13 at 12:32
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I think the norm of the OP is $E\otimes^2_{\mathbb R} \mathbb C$ for the $\ell^2$-tensor product norm (i.e., the one such that the tensor product would be the space of $\mathbb R$-linear Hilbert-Schmidt operators $\mathbb C\to E$). Thus it should be compatible with duality. The injective norm would be the one induced from $L_{\mathbb R}(\mathbb C, E)$ with the operator norm. –  Peter Michor Feb 11 '13 at 15:37
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2 Answers 2

Not an answer, but this is a bit long for a comment.

A Banach space is a dual space iff there is a total family of continuous linear functionals so that the unit ball of the space is compact in the weak topology on the space generated by the family of functionals. From this it is easy to see that if $E_\Bbb{C}$ is a dual space, then $E_\Bbb{C}$ is a dual space when considered as a real Banach space, which implies that there is a norm on $E\oplus E$ which is a dual norm and the projections onto the copies of $E$ have norm one.

This suggests the following questions, which as far as I know are open problems.

  1. If $E\oplus E$ is isomorphic to a dual space, is $E$ isomorphic to a dual space? This question is equivalent to: if $E_\Bbb{C}$ is isomorphic to a dual space, is $E$ isomorphic to a dual space?

  2. Same as (1), but with the additional condition that $E$ be separable.

  3. Is every complemented subspace of a separable dual space isomorphic to a dual space?

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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 $E\times E$.

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