Please, have a look at the following paper:

Plichko, A.N. Condition of conjugacy of WCG-spaces. (English)

Math. Notes 23, 152-153 (1978).

It contains a result of the type you are searching for. The exact statement you can find in Zentralblatt review to the original Russian version of the paper: Zbl 0417.46017.

The class of WCG spaces contains in particular all separable Banach spaces, so the result is quite general.

All the best, Vladimir.

Thank you very much. But I would like to ask, wether the space B(H) of all bounded linear operators on an infintie dimensional Hilbert space fits into this setting.

Best regards Manfred

B(H) is not a WCG space, so it does not fit into Plichko´s theorem. But B(H) is a dual space, because there is a locally convex Hausdorff topology \tau on B(H) that is weaker than the norm topology, and such that the unit ball of B(H) is \tau-compact.

This topology \tau is the topology of weak pointwise convergence.

Best regards, Vladimir

Thank you for your answer. The (unique) predual is knien to be the space of trace class operators. My problem is concerned with C*algebras: are there useful criteria to decide Werther such one is a W*algebra. Best regards Manfred

Is there any example of a C*-subalgebra in B(H) that is not w*-closed in B(H), but is a dual space to some Banach space?

Some characterizations of conjugate spaces, sufficient conditions and examples can be found in  "Geometric Functional Analysis and its Applications", by R. B. Holmes, section 23, pages 211-221. The main theorem is that stated at page 213 (Dixmier-Goldberg-Ruston theorem). Passing over these characterizations, I would like to recall an interesting class of conjugate spaces, mentioned in the same book. Let X, Y be Banach spaces and consider the space B=B(X, Y*) of all bounded (linear) operators applying X into Y*. Then any such operator - space is a conjugate space. Concerning reflexive Banach spaces, it is obvious that such a space is conjugate. On the other hand, on every non-reflexive Banach space X there is an equivalent norm such that  X so normed is not congruent to a conjugate space.