TITLE: Preduals of B(E)
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: As Yemon mentioned ages ago (sorry!) I explored this a bit in http://arxiv.org/abs/math.FA/0604372
We say that a Banach algebra $A$ is a dual Banach algebra if $A$ is isomorphic to $E^*$ for some Banach space $E$, such that the multiplication in $A$ becomes separately weak$^*$-continuous.  If $X$ is a dual space, then $B(X)$ is the dual of $N(X_*)$ but a little calculation shows that the multiplication is only weak$^*$-continuous on one side.  To get a dual Banach algebra, you need $X$ to be reflexive.
My little result is that if $X$ is reflexive, and also has the approximation property, then $N(X)$ is the unique dual Banach algebra predual of $B(X)$.  To be precise, if $B$ is another dual Banach algebra, and $\theta:B(X)\rightarrow B$ is a linear bijection, is bounded, and is an algebra homomorphism, then $\theta$ is necessarily weak$^*$-continuous (so I don't need to assume that $\theta$ is an isometry).