Banach space which has Radon-Nikodym Property +Weakly sequentially complete but not reflexive.

Question

I have been trying to understand reflexivity of a Banach space from geometric point of view. I know that a reflexive space has Radon-Nikodym Property (RNP) and is Weakly sequentially complete(WSC) by default. Then there are Banach spaces which have RNP but are not reflexive. Similarly there are Banach spaces which are WSC but not reflexive. $\ell^1$ has both RNP and WSC. Are there any other non-reflexive spaces which have both RNP and WSC?

Answer 1

A good candidate of a non-reflexive WSC Banach space with RNP is the dual $X^*$ of $X$ with property (V) that contains no copy of $\ell^1$.

If $X$ has property (V), then $X^*$ is WSC. $X^*$ has RNP iff every separable subspace of $X$ has a separable dual. Thus, if $X^*$ has RNP, then $X$ contains no subspace isomorphic to $\ell^1$. However, there are spaces $B$ that contain no copy of $\ell^1$ and $B^*$ does not have RNP (James tree space is an example as far as I can remember).

For example, let $A$ be a scattered $C^*$-algebra, a $C^*$ algebra in which the spectrum of every self-adjoint $x\in A$ is at most countable. A $C^*$-algebra is scattered iff it contains no subspace isomorphic to $\ell^1$. Its dual $A^*$ is nonreflexive, WSC, and has RNP.

For a particular example, for a compact group $G$, the group $C^*$-algebra $C^*(G)$ is scattered. Its dual $B(G)$, the Fourier-Stieltjes algebra of $G$, is nonreflexive, WSC, and has RNP.

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Preduals of purely atomic von Neumann algebras are also non-reflexive, WSC, and has RNP. The Fourier algebra $A(G)$ of a compact group $G$ is a particular example.

Answer 2

• Separable dual spaces have the RNP.

Also,

• the projective tensor product of $L_p$ and a WSC space is WSC,

which gives you plenty of examples.

In particular, the predual of the space of bounded linear operators on $L_p$ ($1<p<\infty$) is an example of the kind you seek (it is the dual of the space of compact operators on $L_p$).