indecomposable dual Banach spaces

[Volker Runde]

A Banach space E is called indecomposable if there are no infinite-dimensional subspaces X and Y of E such that E is the direct sum of X and Y.

There are examples of reflexive indecomposable Banach spaces.

My question is: Is there a indecomposable Banach space that is a dual space, but not reflexive?

Any pertinent hints will be appreciated.

Volker Runde.

[Philip Brooker]

Any (non-reflexive), quasi-reflexive indecomposable space is such a
space. The existence of such spaces is shown in, e.g., the book Ramsey
Methods in Analysis by Argyros and Todorcevic (see in particular
Theorem V.1). Since every quasi-reflexive space is a dual space by a
classical result of Civin and Yood (Quasi-reflexive spaces, Proc.
Amer. Math. Soc. 8 (1957), 906--911), the desired example is achieved.

See also my remarks in the comments below my answer to a Mathoverflow
question at http://mathoverflow.net/questions/46138/does-taking-the-dual-space-stabilize/46191#46191
for slightly more information.