For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.

If $E$ is a Banach space then is it known whether

$B(E)$ is always a dual Banach algebra?

The predual is always unique?

I'm aware that 2 can fail in the case of a general `dual Banach algebra', so if the answer is "No!" can we place appropriate conditions on $E$ to ensure that 1 and 2 hold? If this is well-known then appropriate references would be useful.

`$E\hat{\otimes} E^*$`

, and this is unique as an isometric DBA predual; if E also has the AP then I think Daws's results imply this predual is unique as a DBA predual, but I may be misreading. – Yemon Choi Jul 3 '11 at 4:04